Using a grout saw or an electric tool such as a Dremel with an appropriate attachment, begin to remove the grout from between the marble tiles. As you work, be very careful to avoid the edges of the tiles themselves, as you do not want to chip or scratch them. This can kick up a lot of grout dust, so make sure the area is well ventilated. You may also want to wear a breathing filter.
If you just want to do a quick job you can remove the grout down just the middle of the joints between the individual pieces of marble. This will allow you to finish the job quickly, and will look somewhat better. However you should strive to remove all of the grout as thoroughly as possible if you want the best look and the most consistent coloring.