1. Scrub dirty grout using warm water and a medium-bristle brush.
If you don’t already have a grout scrubber, most home centers and hardware stores carry a number of products that are specifically designed for the purpose of cleaning tile grout. To avoid damaging the grout, opt for a medium-bristle nylon brush, not a hard steel one. Simply spray warm water on the grout lines and scrub in a circular motion, then let it dry. Don’t use too much water or let it sit on the grout for too long. Remember: Porous cement grouts absorb water, which could lead to mildew.
2. Spray the grout with equal parts vinegar and warm water.
If you know your grout has been sealed, but it has accumulated heavy dirt or mild stains, turn to vinegar, that trusty old household staple. Fill a spray bottle with a half-and-half solution of vinegar and warm water. Spray the mixture on the grout, let it stand for 5 minutes, then scrub the surface with a stiff brush. Avoid using vinegar on unsealed grout.
3. Apply a baking soda paste and then spray with vinegar.
Cleaning grout with baking soda will bring even more power to the party. Here’s what to do: Cover grout lines with a paste of baking soda and water, then spray on the vinegar solution listed above (remember, only apply vinegar if the grout has been sealed). Once the mixture stops foaming, scrub with a brush, rinse with plain water, and wipe dry. If the grout is unsealed or needs resealing, apply just the baking soda solution and scrub carefully.
Source: Bob Vila