Caution--most of these might also remove the grid lines.
Simple Green--careful, the stuff is powerful and should be diluted.
Mess Master (florist's chemical).
Draw on the permanent line with something you know you can erase, then erase *that* as normal. Sometimes the permanent line comes with it.
Rubbing alcohol, a soft toothbrush, and a strong arm.
Mr. Clean Magic Erasers have a good rep for this.
Tearful supplications to the Mat God, accompanied by cruel sacrifices of metal d4s (his greatest foe).