I like music with good instrumentation (I don't know if that's the right word). That is, music that cares about all the sounds and melodies and stuff and not necessarily the lyrics. It's not the only thing I care about, and I certainly like some music with simple, repetitive beats, but I really dig the kind of music that, if you took the lyrics away, would still be good to listen to.
But lemme just thrown some names and things out there: Kate Bush, David Bowie, Imogen Heap, Of Monsters And Men, Florence + the Machine - lots of stuff in and around the rock genre. And lots of movie soundtracks, classical music, and that sort of stuff. And lots of Japanese music.