Wipe back and forth over the chalkboard with a dust mop. Start at one of the upper corners and go back and forth, horizontally, working your way down as you wipe off the chalk dust residue. By starting at the top, dust will continue to fall downward, allowing you to wipe it up with the dust mop.
Dampen a chamois in a solution made with 1/4 cup of white vinegar and 1 gallon of water. Soak the chamois in the diluted vinegar then wring out the excess moisture.
Wipe the chalkboard, starting at the top and working from one side to the other. As the chalk dust collects on the cloth, rinse off the chamois and apply more diluted vinegar solution.
Dry the chalkboard with a lint-free cloth.