are you talking about a 'self-healing' rotary cutter mat? or a kitchen cutting board?
fine grain sand paper will clean up your self-healing mat (might work on a cutting board too- depends on what it is make out of.
also a 'green pad' (scrubby) or a magic eraser may clean up your mat...did you use a rotary cutter? or some other tool? i've always used scissors to cut fleece- but have had a problem like you describe from cutting batting squares.