What is a good wedding song? Jam bands?

I am getting married in july and I am trying to come up with a first dance song. We like rock music. Some of our favorite bands are The Dead, Phish, Dylan, Beatles, Hendrix, and a few newer jam bands string cheese, widespread, so on and so forth. We want something that reflects our style in music that is slow and romantic. That is really hard seeing as though none of the music we listen to is slow or romantic really. Any ideas for a song in this general area of music?