It is best to use a grout sponge because regular sponges may drag too much of the grout out of the joints of the tiles. Grout sponges are fairly inexpensive, so it’s best to pick one up for the best quality job. Make sure your sponge is just damp and not soaking wet or you will get your grout lines too wet and it will start dripping down your tile.
You can use a damp grout sponge to get most of the grout off the tile and then follow up with a wet rag to get the rest of the grout residue off your tile. Be sure to rinse your rag and sponge in a bucket of water frequently to avoid leaving behind too much grout residue on your tile. Grout residue can be hard to remove if you let it dry for too long.
Check out our article What Is Grout Haze? and how to avoid it before you start your next tile job!