Scrub the shellfish with a stiff brush under cool running water. If the mussels have beards, pull them off. Pat dry.
Prepare a medium-high gas or charcoal fire. When the grill is ready, set the shellfish directly on the cooking grate (oysters cupped side down). Grill until the shells pop open, the meat is plump, and the juices are boiling and sizzling in the shell, 3 to 7 minutes for mussels, 6 to 10 minutes for clams, and 4 to 6 minutes for oysters; there’s no need to turn the shellfish. (Cooking times can vary widely, because the larger the shellfish, the longer it will take to cook; on the other hand, the fresher it is, the shorter the cooking time.) Remove the shellfish as they finish cooking, protecting your hands with tongs or hot pads. Discard any that don’t open. Serve on the half shell with the dipping sauce of your choice.