The best thing you can do to keep your grout clean is to seal it. Grout is porous, which means it will absorb dirt and spills. But a grout sealer forms a protective barrier on top to help keep it clean. You can find liquid grout sealers at most hardware stores; you just apply the liquid on the grout lines and it soaks in and seals it. You can also use a spray solution of 1 part vinegar to 1 part water for regular cleaning and maintenance after you seal the grout.