You can get good results using Simple Green soap at a 50/50 mix with water. Brush on the Simple Green solution with a very soft good size width paint brush that won't abrade the grille cloth fibers. Work it into the grille cloth gently, then let it sit for awhile. I rinse my projects off in the shower with a gentle warm spray. Repeat if necessary. Let dry over night. Don't apply any heat, hair dryer, etc., just let dry naturally.
Good Luck