Scrub the grout clean with household tile cleaner and a stiff toothbrush, taking up all surface dirt. Rinse off the grout. Let it dry.
Put 1/2 cup of oxygen bleach in a bowl. Add hot water while stirring with a spoon. Add the water slowly, making a paste that's about the consistency of cake frosting.
Spread the paste over the grout lines, using the rubber spatula. Put it on thickly, so it covers completely. Mix up more paste if necessary to cover it all.
Allow the paste to sit on the grout for half an hour. It bubbles and fizzes as it works into the grout surface and lifts out stains. Watch the lines and add more paste if any areas absorb the paste completely and become dry.
Scrub the grout lines vigorously with the toothbrush. Rinse well. If the grout isn't as clean as you want it, repeat the whole process until it is.