By default, the Amazon MP3 Downloader application will automatically add downloaded files to iTunes for you. But, also by default, it doesn’t clean up those downloaded files once they’ve been copied into your iTunes library; they just sit there in your Amazon MP3 folder until you go back and delete them. You can correct this with a little tweak to the Downloader's settings.
In the General tab of the Amazon MP3 Downloader app’s preferences, open the Save Downloads To drop-down and navigate to your Automatically Add to iTunes folder (in your iTunes Music folder by default). While still in the General tab, uncheck Add Downloaded Tracks to iTunes.
Next time you purchase an MP3 from Amazon and it finishes downloading, click on View Download Folder on the Downloads Complete dialog. You’ll see that there are no files in your newly designated download folder. (You can select Don’t Show This Message Again, if you wish to skip this now-redundant dialog in future.) If you open iTunes, you should see your new tunes, ready to play.
Thanks to Mac OS X Hints reader pgor for tipping us off to this one.