I have seen some discussion about Rhythmbox not having an Amazon plugin & Banshee not working with Amazon in 12.04.
So here is the solution:
sudo apt-get install clamz
Then go to the clamz download page here:
and choose your country's link at the bottom of the page.
Then to test go to Amazon and download a free mp3.
After you have checked out it will offer to save a file. (.amz file)
Save to your Desktop.
Right click the file and open with Clammp3 downloader.
A terminal opens and your music will be saved to your Home folder.
Handy little tool