Do you have FLAC files, as opposed to the regular CD's? I don't see why you'd have a problem with loading a regular CD into an mp3 player. Mine does it without a problem. But I have some boots in FLAC format that won't load.
I haven't owned an actual CD player for a few years now. When my last one died, I just hooked up my old Sony DVD player to the receiver, and it plays just like a normal CD player. I upgraded to a better DVD player, so I just figured I'd try to save money by seeing if my old DVD player would do the job.