We have clean tile and grout! Thanks to all of your helpful suggestions a couple weeks back, I'm here to report that with a little time and elbow grease I got our guest bath tile floor sparkling clean again.
A few people suggested Barkeeper's Friend. I already had some in the house, so that's what I tried first. Did you know it also makes stainless steel look like new???
I made a slurry of Barkeeper's Friend and water. Working in small sections, I poured it over the floor and let it sit for a few minutes. Then using a stiff bristled brush, I worked it into the grout and rinsed it off with a clean rag. It worked like a charm!