Choose a scrubbing solution. Grout, especially between floor tiles, gets especially dingy and dirty over time. Depending on the severity of your grout discoloration, you will need to use a different cleaning solution. For mild discoloration, use a combination of baking soda and hydrogen peroxide to form a paste. For more severe discoloration, use an oxygen bleach to whiten your grout.
Do a preliminary clean. In order to avoid extra work when you really start scrubbing, do a mild clean down of your grout prior to deep-cleaning it. Use a mixture of bleach and water to kill mold and mildew, and wipe off any grime or dirt that might be present on the surface.
Apply your cleaner. Working in small sections of tile/grout (try 1sq. ft. at a time), apply your cleaner to the grout. Leave it to set for 3-5 minutes, as this will make the scrubbing much easier.
Start scrubbing the grout. Use a brand new toothbrush (electric is preferable) to scrub away the dirt and discoloration on the grout. This can be relatively time consuming, so don’t give up if at first you don’t succeed. Use fresh water and a rag to wipe off the cleaning residue, and apply another coat of your cleaner if necessary.
Continue cleaning your grout. Work your way outwards from your starting point, using the aforementioned process. Add cleaner to small sections, let it set, and scrub away until bright, clean, and shiny grout can be seen underneath.