Anime-Planet and MyAnimeList have user submitted suggestions for shows on a "if you liked X, you might like Y" setup.
As a developer myself, this is something I've wanted to look into making, and MyAnimeList seems to be the most common site to be used for an engine as they provide an API to programmers.
There are a couple of engines based off of MyAnimeList that people have made such as AnimeRecs (which is down at the moment) or mal.oko.im
There's also a new anime cataloguer Hummingbird which gives you a range of recommendations based on what you've watched, what genres you like, which are your favourites. It seems to be the easiest to access and isn't soley show-specific.
Other websites which provide anime have recommendations engine built in, Usually these are based off of what similarly interested people have enjoyed also.
Crunchyroll (Thanks @David_Starkey ):