I wanted to be mindful of spoilers, and I do want the whole crazy, tangled web of responsibility untangled, but more specifically, I wanted to know if the Third Hokage knew about Itachi's mission, and whether he was the one who ordered the massacre. The show really seems to tiptoe around the question of how involved good old Hiruzen was, so I'm curious if there's any definitive answer.
Original answer, using Kakashi: Shadow of the ANBU Black Ops version of events
Unless I'm mistaken, whenever the issue of blame comes up in dialogue, it is always stated to be the village elders and the Third Hokage who ordered the mission. In other words, the Third Hokage is always grouped in there. I think this actually comes up several times, but I'm working with my memory here (I'd like to get specific references if I can find them, but this is a long series).
The Uchiha massacre is revisited many times throughout the series, but for specific details, I believe the flashback arc Kakashi: Shadow of the Anbu Black Ops, is the best option. There are several episodes within this arc pertaining to Itachi and the massacre, and I think this whole arc does a really great job of tying up a lot of otherwise confusing plot points.
So, let's start with who planned the massacre. Right off the bat, I think it's pretty obvious Danzo, the Third Hokage, and the two village elders all had a hand in preparations. This comes from Episode 359 – Kakashi: Shadow of the ANBU Black Ops – The Night of the Tragedy. While they're arguing, Hiruzen says, "I won't let the Uchiha stage a coup." and the village elders take a wait-and-see approach, saying that's why they have Itachi. In another conversation, after it becomes apparent that the Uchiha are definitely staging a coup, Hiruzen tells Danzo, "Preparations only."
Once Hiruzen has said that, it's very apparent that Itachi's mission was planned well in advance. It's a last-resort tactic (at least in Hiruzen's opinion), but very much per-meditated. Further proof: when Hiruzen arrives at the scene of the massacre, he immediately reprimands Danzo. In other words, he knew exactly what happened.
This episode makes it clear that Danzo is the one who actually pulled the trigger on this loaded gun, but there's actually subtle nods that Hiruzen either authorized or ordered Itachi's mission as well. Notably, Itachi also seems more loyal to the Third, not Danzo. See the contrast: after his mission, he threatens Danzo, but with the Third, he kneels before him and asks for his help protecting Sasuke. Also notably, the Third Hokage doesn't rebuke Itachi at all. Instead, he thanks him and says it was regrettable that this was the only solution. In other words, after Danzo ordered the mission, Itachi and the Third likely discussed the problem and came to the same conclusion that the Uchiha needed to be stopped (this is all conjecture, of course, but I believe it's supported by the events shown). Though it's unclear who convinced whom, I suspect it was Itachi, who saw there was no other way to stop the Uchiha, who convinced Hiruzen, given how reluctant he always was.
Now, another reason I suspect this is how events played out, is because of what happens to Danzo. This whole arc has been about Danzo pulling the most infuriating stunts behind the Third Hokage's back and always getting the upper hand. But here, it's reversed. Are we expected to believe someone as experienced as the Third Hokage would continue to let this happen? That Danzo just went ahead and pulled the trigger on the Uchiha massacre behind his back, yet again? No, rather, the Third approved Itachi's mission and then made Danzo take the fall. According to his design, he then proceeded to use that as an excuse to strip Danzo of power. In this way, he finally gets the upper hand on Danzo. I may just be giving the writers too much credit here, but I actually think getting soundly beaten here very neatly explains why we don't hear anything from Danzo during the original Naruto series (As previously stated, I honestly think this little arc neatly explains and ties together so much of the series).
If you still don't believe Hiruzen was responsible for the massacre, I have one last quote, from episode 366: The All-Knowing. Sasuke directly asks Hiruzen about the massacre, and Hiruzen actually seems to confirm he ordered it: Hiruzen says, "I not only had him kill his brethren and bear the false charge of traitor. But also, keep tabs on the Akatsuki by himself."
I think the show tries very hard to keep Hiruzen's involvement deniable, so you can form your own conclusion if you want, but to me it seems pretty apparent given the evidence that he's guilty, and probably one of the most guilty.
We shouldn't discount Itachi's involvement either. Not only is he the one who actually carried out the murders, but Itachi is very much looking for the best solution on his own, and the Third Hokage later states that even at the age of seven Itachi had the mindset of a Hokage (this was after being resurrected by Sasuke via Orochimaru's jutsu). He is not an idealist like Shisui, and ultimately prioritized the village above his clan. I very much believe Itachi was an active participant and conspirator in the Uchiha massacre, and not just some pawn who went along with what Hiruzen and Danzo told him. Itachi tactfully played his cards as best as he could, but could not prevent the Uchiha from revolting. The best he could do was eliminate them as a threat to the village, and threaten Danzo so he could keep Sasuke safe.
Lastly, there's Tobi to consider.
(AKA Obito Uchiha)
He had basically nothing to do with planning the mission, but was rather simply an ally recruited by Itachi. His only real motivation for helping was to gather a lot of Sharingan. He was an opportunist, and Itachi of course wanted to use him in return.
Update via the Itachi's Story light novels
The version of the Uchiha massacre and preceding events in the Itachi light novels contains many similarities to the version in the Kakashi: Shadow of the ANBU Black Ops episodes, however, there are differences, and some things are more clearly explained in the books. This has forced me to reevaluate blame in some regards. However, as these seem to be two different versions, I will retain the previous answer above. Note: these books have been adapted into the anime, but I have not seen these episodes yet.
Tobi: It isn't really explained why Tobi is popping into the village in the previous version, but this novelization tells us he's been meeting with Yashiro Uchiha and others, likely in order to agitate the Uchiha and push them toward a coup. When Itachi asks if he's trying to instigate a coup, Tobi sidesteps and says he simply, "Taught them a little something about their clan." But in spite of his denial, we can be pretty certain Tobi was up to no good, especially since Yashiro represents the most radical bunch of the Uchihas. Thus, Tobi is far more responsible for the fate of the Uchiha than initially thought. Not only did he help kill them, but he helped bring about the circumstances that made them a threat to the village.
Danzo: Speaking of instigating, this novel puts us inside Danzo's head many times, and what's in there isn't good. He often thinks about his "long-held ambition", and just in case you haven't figured it out after the first time he mentions it (it's pretty obvious), the book makes it clear that is to destroy all the Uchiha. From page chapter 6, page 150:
The coup in ten days. The time had come at last. He would finally be able to remove the biggest obstacle threatening the security of the village. Danzo trembled with glee at being able to personally put an end to the history of enmity that went back to the founding of Konoha.
From the moment Danzo discovers the gifted Itachi, he thinks of nothing else but maneuvering him into killing his own clan. The perfect scenario. There really isn't any doubt that Danzo was egging the clan on as much as he could from the shadows, pushing them towards a coup. Overtly, he even slashed their budget. This book makes it clear that Danzo is the one who leaked Naruto's connection to the Nine-Tails to everyone, thus turning dissatisfaction away from the village and onto him, a perfect little scapegoat. This is the kind of stuff he pulls all the time, so of course he's at least partly responsible for the village's animosity towards the Uchiha. It is highly likely that he even killed Shisui to prevent things from being mediated (Danzo's rhetoric is pretty good in this book though, much improved from the series, and I do believe it is probable that turning Fugaku over to their side wouldn't have prevented the coup, since it was being pushed by more radical members anyway). His perfect little plan requires an Uchiha to kill the rest of his clan. It could lead to animosity and anxiety amongst the other clans if Anbu did the deed, as they would wonder if they're next. But not so if a deranged Uchiha slaughters his clan. So in this version of events, Danzo is even more responsible. This whole massacre was premeditated for years, and possibly decades.
Itachi: Danzo's premeditation is good news for Itachi's share of the blame. In this version, Itachi is maneuvered into a position where he really has no choice but to carry out Danzo's dark wish. However, the reason he's able to be maneuvered thusly is because of how mature he is, and how well he understands the situation. At one point, he tells the Uchiha dissenters that if they fight, they will lose. And when he sees their plan to kidnap the Hokage, he notes how absurd it is. Itachi knows that if the Uchiha perform a coup, they'll get wiped out anyway, and they'll take the village with them, because the civil war would weaken the village, and that weakness would invite a new ninja war. Itachi accepts Danzo's mission, because he understands sacrificing the Uchiha is the only available way to prevent disaster, and to at least save the village.
Unfortunately, what's also clear about Itachi in this version is that he really does harbor a lot of hatred for his own clan. He gets really irritated at them not seeing the big picture, and during the confrontation over Shisui's death, he throws shuriken at the clan crest and is prepared to kill (but no in front of Sasuke). Some Uchiha he gives a peaceful death, notably Izumi, but he puts his all into torturing Yashiro, one of the worst instigators.
Third Hokage: Hiruzen Sarutobi's main sin in this version is incompetence. There's no getting around it: he's portrayed as a complete fool. Strangely, Itachi doesn't even seem to like him, and views him as incompetent, which seems to contradict the other version of events. Itachi wonders how seriously he thought about the Uchiha, and why he never met with Fugaku. The story doesn't go into Hiruzen's perspective much, but if he had anything to do with the Uchiha massacre, he sure managed to convince Itachi otherwise by his seeming incompetence. He was Hokage when this happened, so as far as I'm concerned, there are only two possibilities: either he's a complete fool, or he's complicit and approved it. The show really tries to save his image as the good guy and sugarcoat it, and it just tries to not answer the question of his involvement, which is infuriating to me. The Itachi books go the incompetent route. And yet, in spite of the fact that Danzo probably told him nothing, Hiruzen is still quick to strip him of his position after the massacre, just in the other version (while also noting that this might not stop Danzo from doing things in secret).
From chapter 6, page 148-149:
"But Hiruzen," [Danzo] said, quietly. "The Uchiha clan can no longer be stopped. In which case, in order to avoid chaos, we must strike as soon as possible... That also means the children, who know nothing of this plot."
"That is not something to be said in front of Itachi!" Hiruzen snapped, as Itachi simply watched. [...]
On the surface, Hiruzen's words could have been taken as a kindness, but hidden in them was also proof that the Hokage was thinking the same thing as Danzo. Hiding ugly things from people's eyes like this was the true nature of this village.
Exactly how seriously has he thought about the Uchiha?
Itachi had his doubts about Hiruzen. Despite his concerns of civil war, he hadn't once come to speak directly with Fugaku and the others. And it was Hiruzen who had removed Shisui from his regular duties, and allowed him to maneuver in secret. On the surface, he spoke as if from the heart, but in making Itachi independent as a team leader, so he could continuously monitor the clan, Hiruzen was doing the same thing as Danzo.
Itachi compares Hiruzen to Danzo twice here. Read into that whatever you want. What's clear is that Itachi does not have a favorable opinion of Hiruzen.
The other two councilors: They don't really make their own decisions. Basically, they just do what Danzo tells them. They approved of slashing the Uchiha's budget. At least in this version, I kind of doubt Danzo would consult with them about his plan of massacring the Uchiha. If he did, they'd probably approve since it would stop the coup.