I know you're asking about cleaning the tile but to be sure of the 'stage' of the project, there a need to clarify first.
You 'drag' the wet trowel at a diagonal to apply the grout (not clean it as you may, or may not, be doing.) Once the tile is in place, you drag a damp sponge diagonally across the tiles to clean off the tile (which you may, or may not, be doing.) No biggie as you know the 'stage' of the job it is you're at.
If you'd like to read more, I suggest you take a quick peek at Hometime Projects. Click PROJECTS, then click Text Only Project List, then click Ceramic Tile, then click Grouting Tile. It's a 2 minute read so for peace of mind, check it out.