My mat is the same vinyl as a chessex, but i drew the lines myself, so when I clean it, I have to make sure NOT to take off the sharpie. Red never touches my mat, but my blue Lumocolor marker is my trusted friend.
I do, however, wind up with a thin layer of blue all over it, though, after I do a lot of marking and wiping. Windex multisurface took off even residue I couldn't see. But it's a vinegar formula, and if you've tried that already, you might have limited success.
As far as those suggesting buying a new mat, my two cent are: don't underestimate the goodness of making your own. My mat is a full 36x45(at least) inches, and it only cost me five bucks, plus the price of a new sharpie and an hour with a yardstick.