For some stains, simple household items like vinegar or baking soda can be effective cleaners. Use vinegar diluted with an equal amount of water, or make a paste of baking soda and a bit of water to clean the grout with. Spray or dab the cleaner onto a small length of grout and scrub with a stiff brush. Scrubbing in small circles rather than straight up and down will remove more debris from the grout. Rinse well with water and mop up and remaining moisture with a sponge or paper towels.