I'm a Rhapsody user and there are issues getting the Rhapsody app to play in the background. I'll start the app, play a song, hit the phones home button to send Rhapsody to the background. Then I'll move on to other things, Facebook, Reader, and then Rhapsody will just stop.
I've brought the issue up to Rhapsody, just letting others know.
Google Play Music acts as expected, never closes when in the background.