I like DW256's answer. However, it appears that you want a word that doesn't include "grid", so . . .
If you just want to refer to the pattern, you could call it "reticulated" or "reticulate":
resembling a net or network
especially : having veins, fibers, or lines crossing
Perhaps the most specific adjective for "this device has gradations marked on it" (that could also describe rulers, measuring cylinders) would be "graded" or "graduated"1:
marked with divisions indicating degrees or units of measurement
a graduated cylinder/tube
graduated measuring cups
That being said, I've never heard either of these terms applied to cutting mats. DW256's suggestion (or something similar) would be much more natural.
1 If it only had lines without gradations, then alphabet's suggestion of "ruled" would work better.