Summarize:

Dampen a clean cloth or a sponge with warm water and add 1 to 2 drops of mild dish soap. Then, gently wipe the placemats with the cloth to remove any dirt or food residue. Rinse the placements with water once you are done cleaning them and let them air dry. Do not rub or scrub the placemats, as this can damage the bamboo or wicker. Gentle wiping is best for these types of placemats. If you have placemats made of cork, white vinegar is a good cleaning agent. Mix 1 part white vinegar with 1 part water. Then, dampen a clean cloth with the solution and wipe down the cork placemats. You can also put the solution in a spray bottle and apply it to the placemats in a few sprays.  Cork can mold if water is left sitting on it. To avoid this, use a clean cloth to absorb any water remaining on the cork once you clean it. Press the cork dry using the cloth. Then, let it air dry overnight.  You can also use sandpaper to gently buff out marks or stains on your cork placemats. These materials do not hold up well in the washing machine or the dryer. Use clean cloths and sponges to clean these types of placemats. They do best with a gentle hand cleaning once a week.
Wipe bamboo and wicker placemats with soap and water. Use white vinegar on cork placemats. Do not put bamboo, wicker, or cork placemats in the washer.