Well, I can only play them in a very roundabout way.
I'm running LMDE 32-bit, and whenever I insert an audio CD it does seem to be mounted ("Audio Disc" icon on the desktop). If I open the CD in Nautilus, I'm prompted to open Rhythmbox, which is where the problems come in.
If I open Rhythmbox and tell it to play the CD, it promptly crashes with no error messages or anything descriptive at all. Exaile is a similar story.
If I open Totem, and tell it to play the CD, I get "The playback of this movie requires a Audio CD source plugin which is not installed." I have no idea what plugin that could be - as far as I can tell all the gstreamer plugins are installed, just looking at Synaptic.
Lastly, I can get CDs to play in VLC, but not from Nautilus. If I try right click > open with > VLC, I get "VLC is unable to open the MRL 'cdda://sr0/Track%201.wav'. Check the log for details." However, if I open VLC first, select Open Disc, select Audio CD and press play, I finally get playback (after applying this fix: http://linuxbsdos.com/forum/thread-61.html
Obviously something's wrong here, as I used to be able to play CDs on my old Mint 9 installation. Any ideas?