You don't tell us what method you're attempting to use when you burn these music CDs, so it's difficult to troubleshoot.
My best guess is that you're copying a bunch of music files onto a CD-ROM disk and it is creating a CD filled with data (music) files, and not creating a playable music CD. Any such data CD will "work fine" in your computer, and allow you to play the music just as you would if the files resided on your local hard disk, but will rarely play in an external (music) CD player. (I say 'rarely', because there are some CD players that will recognize and play a collection of (for instance) MP3 files written to a data CD/DVD disk, but those sorts of players are relatively rare.)
The application DeVeDe (available in the repositories, and attainable through Synaptic, apt-get, the Software Manager, etc) will create listenable music CDs and watchable video DVDs, with very little effort.
need I say more??