Dirty grout is a common problem, especially when it comes to the bathroom floor. Grout is typically a lighter colour than the surrounding tile, so while the tile itself may not dull as quickly, the grout regularly becomes soiled and discoloured as residue, soap, lotions, make-up and other elements spill onto the floor. To clean the grout properly, you need to do a bit more than a simple mopping, which will work to clean the tile but not always the grout.
Fill a cleaning bucket with 1 cup oxygen bleach and 1 gallon warm water.
Start in the far corner of the bathroom, dipping either a nylon scrub brush or an old toothbrush into the bleach solution. Scrub the brush over the grout, applying pressure as needed until you remove all build-up. If the floor grout hasn't been cleaned in a while this can take a bit of time and pressure.
Rinse a cleaning rag or cleaning mop under warm water, and wring out the rag or mop. Rub this over your entire bathroom floor to lift all remaining build-up.