This HTTP .MP3 stream uses authentication or timestamps in the URLs, so you may have problems downloading such streams. But you can always use software that captures packets or just get it from the cache of your web-browser:
FAQ: How to record/download/capture/save Adobe Flash .FLV video HTTP stream from embedded player

Replay Media Catcher can perfectly rip HTTP .MP3 streams from awdio.com and you don't even need to use its Web Stream Dumper