I sure hope someone comes up with an answer. We have the same problem. The house has a lot of tile floor and the builder used white grout and no sealer, as far as we can tell. The grout absorbs dirt and is almost uniformly brown now except at the edges where people don't walk. Nothing removes the stain except (I've heard) toilet bowl cleaner with bleach. But who wants to spread that around? The fumes might be toxic used on this scale, and it probably would eat up the grout eventually. I'm not even sure it would be cheaper than a product designed for this purpose.
I will say that many years ago we had a similar problem and I just painted the grout with interior latex paint. If the tile is not porous, you can just slap it on and it sticks to the grout but rubs off the tile, as long as you don't wait too long. This works pretty well if the grout is lower than the tile (usually the case) and you don't have a dog with claws that scratch the paint off here and there. I just kept a jar of it around and touched up as needed. It was a pretty satisfactory solution that was within our budget during the time that we had three kids in college. It held up for years, other than the dog claw problem.