If you have grout that's stained or moldy, you may get lucky and discover that cleaning chemicals can handle the problem. Skip the traditional solutions like vinegar, bleach and baking soda, though—they don't work nearly as well as a powdered oxygenated cleaner mixed into a paste (OxiClean is one brand). Or try commercial grout and mold elimination products from the home center. If chemicals work, great. If they don't, you'll have to remove a layer of the old grout and replace it with new.
After chemicals failed for me, I tried everything else: a grout removal attachment for my rotary tool, a grout removal blade in my reciprocating saw, a pulsating tool with a triangular carbide bit, and a tool that looks like an electric engraver with a chisel tip. All of them either destroyed tiles or were painfully slow. Plus, they all kicked up a gritty dust storm. I finally figured out the easiest and fastest combination for the job—an oscillating tool fitted with a diamond blade (about $10). One source for oscillating tool diamond blades is fitzallblades.com.
You'll need at least two diamond blades depending on the size of the job. You'll also need a scraper blade for your oscillating tool (such as the Dremel MM610 Multi-Max Flexible Scraper available through our affiliation with amazon.com) to scrape away caulk at the inside corners (Photo 1). Next, it's worth fabricating a cooling system for the oscillating tool. I built mine out of inexpensive drip irrigation parts (see “Build a Cooling System” below). Since you'll be working with water, plug the tool into the nearest GFCI-protected outlet. If you don't have one, buy a GFCI-protected extension cord. Don't do this project without GFCI protection.
Next, remove the grout (Photo 2). Your goal isn't to remove all the grout—only about 1/8 in. deep or so. Make several light passes until you reach the preferred depth. That'll be deep enough to embed the new grout. Once you're done grinding, clean the tile and let it dry before applying the new grout.