A Magic Eraser and a little warm water can remove residue from the surface of tiles and could be a simple addition to your regular grout-cleaning routine. In particular, regular wiping with a dampened Magic Eraser sponge can remove some of the gray surface on white grout. For heavier stains, you may want to choose it as your first scrubbing option instead of a brush or soft cloth, then progress to something tougher if needed.