Assuming you’ve tried dish detergent and good old fashioned soap already, the next best option is vinegar (which is also eco-friendly). Use a spray bottle to spray some onto the grout, and work the dirt clean using an old toothbrush. For more stubborn dirt, soak some rags or tissues in vinegar and leave the grout covered overnight. First thing in the morning, remove the tissue and brush the grout clean. Voila!