OK, they are using a Flash player to stream their audio. unfortunately this doesn't tell us much about what they are doing because there are many ways to skin the cat with Flash.
But what I'd recommend, Jim, would be sending then a stereo WAV instead of an MP3 file, and letting them convert it to .swa or MP3 using Flash itself. Also make sure that your browser has the latest version (9, I believe?) of the Flash player installed on it.
But I have to tell you, what I heard playing (Dylan's "Don't Think Twice") is not that awful for web audio; I have heard much, much worse, and listening through the web artifacting, one can tell that the original recording you made sounds quite good.
G.