Tile work is a messy process involving a large amount of mortar that's just liquid enough to splatter or spread in unexpected ways under the weight of your slate tiles. If allowed to dry onto your tile surfaces, the mortar will show readily against the slate colored tiles, with even small amounts of mortar making the slate appear discolored and dirty. Removing the mortar before it dries is the best option, but even dried-on mortar that's hard as stone can be removed with enough effort and the proper materials.
Remove fresh mortar from the tiles with a damp sponge. Wipe the face of the tiles immediately after the mortar touches the face of the tile, and then wipe again in two hours with a clean lint-free cloth to remove any residue left behind from wiping the slate with the sponge.
Dampen a sponge in a small bucket containing a mixture consisting of 1 part vinegar to 1 part water. Use the sponge to wet down mortar that has dried on the slate.
Wait five minutes for the mixture to soak into the mortar, then scrub at the tiles with a stiff bristled nylon brush. Break up the soaked mortar by applying firm strokes with the brush, beginning at the edge of the dried mortar and working over the surface of the mortar to the opposite edge of the material. Use a sponge dipped in clean water to wipe away the broken mortar and to remove the vinegar solution. Repeat the vinegar mixture application multiple times if necessary.
Replace the vinegar mixture with a stone cleaner solution and the nylon brush with a bush containing stiff brass bristles if the vinegar mixture does not work. Apply the cleaner to the mortar on the slate tiles, and allow it to soak into the material for the length of time suggested by the cleaner's manufacturer.
Brush off the loosened mortar with the brass brush. Use lighter strokes than those used with the nylon brush to avoid scratching the surface of the tile. Repeat, if needed, to get rid of all traces of the mortar. Rinse the tiles clean with a damp sponge dipped in clean water.
Attach coarse sandpaper to an electric sander and grind away especially resistant or thick patches of mortar. Use small, circular motions on the mortar until you've removed most traces of the material and reached the slate tile surface. Repeat the steps, beginning with the vinegar mix, until you remove all traces of mortar.