I remember futzing for hours a year ago trying to get the Amazon MP3 Downloader (amazonmp3) working on my Fedora 12 (or 13?) instance. I eventually got it working but it’s a pain because the Fedora 11 binary Amazon provides requires a libboost version that isn’t provided with modern Fedora releases.
Now, however, there’s an opensource implementation: clamz. It’s available in most repositories (Fedora 14 for instance). Opening an .amz file will start clamz and download the music to $HOMEDIR/Music by default (you can customize this and more by editing $HOMEDIR/.clamz/config). In other words: it just works.