Amazon.com requires you to install their software if you want to download purchased content to your computer. However, if you are like me and don't like to install garbage software on your system there is a workaround. Amazon Cloud Player uses Flash to play media files. A flash media/video downloader may be able pinpoint and download the files. I this video I demonstrate this with Flash Video Downloader for Firefox.
-- LINKS --
Firefox: http://www.mozilla.org/
Flash Video Downloader: http://www.flashvideodownloader.org/
FVD Plugin for Firefox: https://addons.mozilla.org/firefox/addon/flash-video-downloader/