To make this project as easy as possible, get your hands on a Dremel rotary tool, which features a rotating head with a speed of thousands of revolutions per minute. For this job, use the Dremel bit number 569, which is designed to cut into stone and similar surfaces without wearing out too quickly. However, if you have a large job to do, you may need multiple bits, as they will gradually wear down, and a dull or worn bit creates more tearing and presents more danger to the tile.
Dremel makes an attachment head for the rotary tool called a grout removal attachment; this is the perfect tool for the job. The attachment features an angled support that holds the drilling bit at an angle so that it can tear easily into the grout without hitting the tile. The attachment also features guides that help you stay in the grout line and avoiding cutting or nicking the tile. Find this attachment, as well as the tool and bits, at hardware and home-improvement stores.
Attach the bit, and then the removal attachment, to the rotary tool. Set the rotary tool between 15,000 to 20,000 rpm. Start at the top of your removal area and work your way down; pulling down gives you better control than pushing up. Work slowly so that you have optimal control and are less likely to slip. The grout attachment features multiple depth settings to allow you to remove all grout if necessary. Even if you want to remove all grout, start with the shallowest setting and make multiple shallow passes, working more deeply into the surface gradually, instead of in a single deep pass. This gives you better control over the removal process and slows down the wear and tear on the bit.
Work slowly to avoid slipping, but also pay attention as you move slowly to any resistance the bit feels in the grout. Older tile installations feature metal tile spacers that are embedded in the grout; if you hit one of these unexpectedly it will send the bit reeling and could cause damage to the tile. If you are removing grout just to remove mold, you only have to go as deep as the mold penetrates the surface. For corners and horizontal areas, you may need to remove the grout attachment to hit different angles; just remember to work slowly and at a shallow depth, even without the attachment present.