Hey, welcome
I know what you're talking about, most of the people use those Windows Media Players for their websites, as you mentioned. However, when I did a search I found another player which might help you; and it seems that you can simply paste the code as you used to do ... but you have to read some documentation before.
It's kinda late ... 1:45am so I don't have the time to search for the Windows Media Player, I might search it when I get up :P Anyways, here you go:
XSPF Web Music Player (FLash)
Tell us if you manage to upload it

Hope it helps.