Grilled oysters are delicious and simple and somewhat primeval: humans have been eating oysters cooked over a fire for a long, long time. They're a staple at Hood Canal get-togethers: Usually they're cooked over an open fire, usually it's raining, and frequently the oysters are clusters just plucked from the closest beach.
The only way you can really mess up a grilled oyster is to have the heat set too high. On a propane grill, keep the direct heat to medium, and consider cooking the oysters over indirect heat with the grill lid down so that it turns into an oven. Over an open fire or briquets the rule of thumb is that if you hear the shells popping, the heat is too hot.
Growing up, we'd just throw the oysters on the grill whole, often cooking clusters instead of singles. If you're cooking oysters this way, be sure to set them on the grate cupped side down with the hinge towards you (mantle pointing away). As the oysters steam open, they may pop and explode, and having the hinge towards you helps direct the oyster shell shrapnel in the opposite direction. (If they're popping and exploding, it's a sign your heat is too high, per Tip #1).
If you're taking the tops off the oysters before you put them on the grill, it's a bit less dangerous. But either way, be sure to procure a waterproof and heatproof glove, or handle them with tongs, because the liquid inside will be scalding hot.
If you're cooking oysters during vibrio season (when the water is warm) be sure to cook them to 140 degrees for 15 seconds to be sure they're safe to eat.
The classic way to serve barbecued oysters is drizzled with garlic butter and splashed with Tabasco. After about 15 minutes or so on a grill, the oysters will steam open about a quarter inch. Pry the shells apart with an oyster knife, and either add a spoonful of sauce if you're eating it right away, or plop the oyster in the sauce if it's gonna be a while.
Nowadays we're using flavored garlic butters, which we add to the oyster before we grill it. We just take the top shell off, add a teaspoon of the butter, and cook everything altogether.
We love Renee Erickson's "snail butter" (available in her cookbook) and Hog Island Oyster Company's chipotle-bourbon oyster butter.
Don't have a grill? Put the pre-shucked oysters on a cookie tray, using rock salt to keep them upright, and bake them at 400 degrees for 8 to 10 minutes.