Instead of removing that grout, I'd try cleaning it first. You may be quite surprised at how well you can clean it. The following is the process I use to clean mildew discoloured wall tile grout. If you follow a similar process on your floor tile grout, you may get it a lot cleaner than you expect. (Then, just replace any grout you can't get clean.)
Clean it with ordinary soap and water and a stiff scrub brush first, just to remove any loose dirt that can easily be removed. Allow to dry.
Now, get some phosphoric acid based toilet bowl cleaner and scrub the grout lines with that stuff and an old tooth brush. Don't bother with anything less than 10 percent phosphoric acid, and try to get one that's at least 15 percent phosphoric acid. Don't use a hydrochloric acid based toilet bowl cleaner on the grout, they're too aggressive and will probably damage the tile, too.
You just want to use the phosphoric acid to dissolve the very surface layer of grout which will probably be plugged up with dirt, soap scum, mildew and whatever else. Once you dissolve that dirt clogged surface layer, rinse the acid off with clean water and allow to dry. (If phosphoric acid is left for TOO long on grout, it will dissolve enough of it to become noticable, so don't leave it on for a long time.) Once it's diluted with rinse water when you're cleaning it up, it'll be too weak to dissolve anything.
Then, after letting the grout dry thoroughly (for several days, even) if there is any mildew growing inside the grout and discoloring it, you can apply bleach straight out of the jug with a small paint brush to the grout lines. The bleach will wick into the grout and kill any mildew inside it. Sometimes, you have to leave the bleach on for hours (or even overnight) for it to penetrate in deep enough.
Finally, apply several coats of an acrylic grout sealer to your grout lines to prevent history from repeating itself. The clear film of sealer over the grout will keep the grout lines easy to clean up with a damp sponge.
And, if there are still spots that you're not happy with, I'd use an inexpensive plastic laminate knife (the kind with the single tungsten carbide tooth at the end of the "blade") to scratch the grout out in those areas you couldn't get clean. I'd scratch the grout out to about the same depth as the grout joint is wide to ensure that the joint is deep enough for the new grout to hold well.