S.O.
I had to clean the grout in a house we moved into. The grout was originally white, but it was actually black, yuck! I rented a grout cleaner (steamer) from Home Depot for 24 hours and went to town. You can buy grout cleaner and put it down first, and then follow up with the grout steamer. It had amazing results! I followed up with a grout sealant after it was all finished. It took me 16 hours to do the whole house but it looked amazing. It was very affordable. Good luck!
The steamer is basically like a pressure washer and it blasts that dirt out of there, it's awesome!