UGH IS RIGHT! I would try (VERY CAREFULLY) with a very sharp knife to remove any loose material in the grout lines and then regrout the whole floor. This has worked for me in the past (when I was a rookie).
You will find that removing "ALL" of the new grout is very difficult. You could use a Dremel or Roto Zip with a tile bit and buzz out all of the grout. But you have to be very careful not to damage the tile doing this. Never mind the complete mess this will make.
If you decide to go this rout, put a good box fan in the window (the dust will be incredible), wear saftey goggles and a dust mask (although I would wear a resperator mask). Work 5 sq ft at a time. Also a good shop vac will come in handy.
If you need further info e mail me at email@example.com.