1) SteamFast steam cleaner (the hand held one)
Simply go over the grout with it, and you should see the muck disappear.
2) Mop the floor with hot water, no soap or cleaner and let it dry.
3) SurfaceGard or a similar sealer on the grout itself, you can apply it with a stiff watercolor paintbrush.
It's time consuming to do all this, but you wouldn't have dirty grout again...at least for a very, very long time.
I had to ask my husband on this one...he was a tilesetter for quite a while.