I did try a fabric eraser that I had on hand already, and it did very little. I can look for the white plastic art type of eraser next time I go shopping. I have nylon scrubbers available...do you use them dry? Or would a little cleaner of some sort work with the scrubbie?
Do cutting mats need cleaning periodically anyway? I have only had mine for a few months, so have not really given that any thought.