Watch any Woody Allen film - some of them don;t even have music over the titles.
There are quite a few songs that don't do background - and the opposite extreme is a 'soundtrack' containing songs that you may have heard on a car stero as someone drove past the actors while they were walking down a street.
Think I read that the new movie 'Cloverfield' has no background music (or very little). I've always been interested to see what a movie without 'muzac' might be like, so will watch it and see.