After soaking, I took out a scrub brush and scrubbed the dirt out. It came right off. And it was pretty disgusting, long, tiring work but so satisfying with stellar results.
I then had to mop and mop and mop all of the black, dirty water along with the grainy/chalky residue left from the Oxiclean. You do not want to know how many giant buckets of black water I dumped out during the cleaning process...from grout alone. Nasty.
It took me many, many hours to scrub the worst parts, including the kitchen and high traffic areas. I have a lot of tile in my house so I broke it into sections. I now have about half of it cleaned and need to tackle one more (giant) area. *sigh* You can see the difference in the grout lines where I stopped. The next problem is removing my rug and huge dining room table. Ugh.
It is so very worth it though. I was sore and tired and practically crippled my wrist after the first go-round, but the floor gleams and looks brand new. I cannot believe it. I now get a little crazy when the kids drip water or try to actually walk on my masterpiece. They may have to live outside now.
My next worry is upkeep. After scrubbing on my hands and knees, I DO NOT want it back to the same state in the very near future. In my research, I discovered that to keep it clean, you put some Oxiclean in the mop water, mop as you regularly would and then leave some of the water to soak for 20-30 minutes. After soaking, rinse with clean water. The oxygen ions are said to cling to the dirt molecules, thereby doing the cleaning (or a good chunk of it) for you. I have yet to see if this really works in the long term (I've tried it once so far) but I will definitely give it a shot. It seems reasonable enough.