(I am new in this "Linux guild" but am actually quite old hat with Byond. This just seemed the most logical place to ask for help in this situation.)
I am running Ubuntu 10.04 and Wine 1.3, and I installed Byond without any trouble.
Everything seems to work pretty well. At first I was in this emulated desktop, and the windows stuck to my cursor if I picked them up and wouldn't be put back down, but when I got rid of the desktop, that problem went away. The only problem that I can see so far, is that I can't interact with popups in-game. This is very problematic, since at least most games I play start with a "New player" popup, and you click a link thingie to enter the game proper.
I have installed the recommended DLLs with Winetricks and everything. So if anyone could shed some light, that would be most appreciated.