I have not run WINE for a long time, but when last I did it would play an old version of WMP but not the then-current version (release 8 IIRC).
As WINE becomes more and more complete in implementing the Windows API, you would certainly think it would play WMP. But I can't say for sure.
Try it and see, or visit winehq.com and see what they have to say about it.
edit: Actually, being curious, I did just that. According to
this page, WMP 10 and 11 don't play though WMP 9 does play with some minor issues. I guess this isn't too surprising; WMP10 has a lot of that digital rights management junk in it and that (I think) would cause it to break in an environment where the real Windows kernel wasn't present.