As steve c said, the grout may have originally been mixed improperly. It's ALSO possible that the floor is 'bouncing/moving' underfoot. If the floor is moving, then the best-of-grout-mixtures isn't going to make a difference. Eventually it, too, will work itself loose. If this is the case, there may be a 'cracked' tile backerboard, NO tile backerboard, damaged or too-widely-spaced floor joists, or some other 'imperfection(s)' under the tile floor. If this is the case, obviously, the floor needs to be stabilized before you can make more permanent repairs. (Unfortunately, if the floor is 'loose' enough, it may require having the job completely re-done ...)
Now, assuming the problem is just a 'bad batch of grout', do as steve suggests. Yes, you need to remove at least 1/8" of grout. There are mechanical as well as hand tools to remove the grout. The only 'problem' you may have w/a mechanical tool is the result of 'inexperience'. If the tool happens to 'grab' something, you may 'skip' over a tile and damage it. (Now you have to replace the tile, and where are you going to get a replacement especially if the tile is 'old'? Perhaps, PERHAPS there is one you can 'steel' that's under a cabinet, or something.) So if you go this route, BE CAREFUL! A hand-held grout removal tool is available for a few $$$s at the Home Center. (I used one in a very small bathroom w/no problems. Yes, it was a bit hard on my hands but it worked out well. If I see your situation, you may be able to remove it in 'chunks')
You can do this yourself or have a Pro do it. It wouldn't hurt to get 3 estimates. I will say that it would be a labor-intensive job. In other words, the only material used is the grout. Most of the Pro's fee is labor. With that, you'll probably get a WIDE range of estimates. You don't necessarily need a Plumber or Tile Pro to do the work (which is who'll charge the most.) BUT, you get what you pay for. Shop around and ask ?'s as to why Pro A is more (or less) than Pro B. Maybe the difference is in the material, maybe it's in the hours spent, who knows - Just ask.
If you do it yourself, get yourself a GOOD book on Kitchen and Bath Renovations. (Time/Life and Reader's Digest, to name a couple, have books written about what you want to do.) It's $$$ well spent! There are some things you need to know BEFORE you start your work. For example, on this form, we've had a couple of DIYers that didn't realize that if the 'grout haze' dried on the tile, it COULDN'T be removed. Sometimes, you wonder why a Pro charges what he/she charges. Experience and Knowledge go a loooooong way. Oh, as for withstanding water and cleaners ... Read up on 'sealing your grout' once you've sponged it in. This will HELP protect your grout. As for 'strong tile cleaners', I dont' think you have a problem there. Tile is a fired-and-glazed material that can stand up to a lot of 'abuse'. If you use STRONG chemicals to clean your floor, consider using LIGHTER strength chemicals but clean more often. (Leave the STRONG ones for the not-so-often uses ...)
My best to ya and hope this helps.
Jay J -Moderator
PS: If you have a Home Warranty still in effect, use it!!!