MP3 Music is possible on the AD-600A and it will play both Constant Bitrate (CBR) and Variable Bitrate (VBR) files. I have tried many different types of media including 74 minute CDR/CDRW, 80 minute CDR/CDRW and 87 minute CDR's all with perfect results.
The AD-600A has a simple on-screen menu having only 8 characters per song name, which can seem confusing at first.

Navigating around an MP3 disc is easy too once you get the hang of it. You use the cursor keys (Up, Down, Left, Right) to move the highlight, ENTER/PLAY to start playing a track, STOP to stop playing a track and the |<< and >>| skip buttons to jump to the next/previous track. If you have multiple folders you can return to the Root Menu by pressing "PROGRAM".
However the AD-600A can have difficulty with some of the earlier firmware if the CD is burnt using the wrong format or character set. If your MP3 CD does not get recognised chances are it has be burnt using ASCII and not ISO9660/Mode 1 standard. If you are using Nero here is how to set up the burn options to get optimal results. Only the two yellow highlighted options are necessary and the other options are up to you but I suggest using ISO Level 2 File Length so you can still read what the MP3 file is if you use the CD in your computer.

It is not all versions of the firmware that need to be burnt like this. Mainly only the early versions like A6 to A10. Anything like A11, N5, GK600, GK600A or GK600B will accept using ASCII character set so you can use that option if you wish.
Also note that GK600B does not cut off the first half second of MP3 music.