You can usually find pre-gridded backsplash tiles at hardware stores. If you can find one you like with small enough tiles, you can have good-looking, durable coasters in minutes.
Instructables user Joekevdv explained that the tile above was cheap at his local scrap store, but you could even grab some at your local home hardware store if you don't have a salvage store nearby. Shave off the loose ends of the grid on the back of the tile, add a little hot glue on the bottom to seal it over, or attach a felt or cork pad, and you're all set. They're easy to clean and look great.
Of course, you may not find this specific type of tile near you, but we're willing to bet you'll find a type of small tile you wouldn't mind using as a coaster the same way.
5 Minute Coasters [Instructables]