Is there a way to exit Front Row and keep my music playing?
Not with Front Row in Mac OS X 10.5.x (Leopard). In Mac OS X 10.4.x (Tiger), Front Row would actually open iTunes in the background to play your music. If you exited Front Row, iTunes would still be running. In Leopard, Front Row has direct access to your iTunes library and iTunes itself is not opened. So, if you quit Front Row you quit the application that's playing the music.
If you want to play music without using Front Row, why not just open iTunes directly? The Apple remote will work directly in iTunes.
If you don't have an Apple remote but have a Wi-Fi network, you could use an iPod touch or an iPhone with the 2.0 software and Apple's "Remote" application to remote control your iTunes. There are also third party remote control devices you might be able to use.
-Doug