Grout cleaning product and a narrow brush on your hands and knees.

I have a steam machine that I use. I steam the grout lines in my floor & you can see the dirt lift up & pool in the water that the machine leaves behind. I use a mop to get the dirt & water up. I rinse my mop after every pass to keep it clean. I do this about once a year & when finished my floor looks new.

I've had decent luck with an oxiclean & water mixture applied to the grout in small areas on my hands and knees with a narrow brush, scrubbing, waiting 5 minutes or so with the oxiclean solution/paste still on, then wiping clean with a rag and water. Repeat this over whole area doing small sections at a time.

Grout does get stained and deteriorated over a period of time if not sealed properly. Some cleaners only offer superficial cleaning assistance, but as grout is porous, you need a penetrating cleaner for a thoroughly sanitized surface. If you find your grout dirty and stained, you can use a deep-soaking penetrating cleaner that enters inside the porous grout and eliminates mold and stains from within. I faced a similar problem in my bathroom tile. I used Imperia Deep Clean to clean my grout properly. Although you get a clean surface after using tile and grout cleaners, you cannot guarantee that staining will not occur again, and so it is important to seal the grout after cleaning.