A clean grout makes a bigger difference than you'd think. Here's the most effective method, plus the mistakes most people make.
Mix baking soda and hydrogen peroxide into a thick paste. Apply it directly to the grout lines using a finger or old toothbrush.
Leave the paste on the grout for 10-15 minutes. For heavily stained grout, leave it for 30 minutes.
Use a stiff grout brush (or old toothbrush) to scrub the grout lines. The baking soda acts as a gentle abrasive while the peroxide bleaches.
Wipe with a damp cloth and inspect the results. For stubborn mold or mildew, apply diluted Soap-Man Power Bleach and let it sit 5 minutes before scrubbing.
Once clean and dry, apply grout sealer to protect against future staining. This makes maintenance much easier.