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Yes you can. I like using tinfoil best. I think putting them directly on the grill dries them out (unless you par cook them). Some ideas...
-Your obvious standard baked potato
-White/yellow potatoes cut in half, seasoned with butter and spices
-"hobo" potatoes - potatoes sliced with onions, garlic, butter and spices fold into a tinfoil packet. cook 20-30 min. over direct heat

Sure. We slice them about 3/8 inch thick, toss with olive oil and fresh herbs, and maybe even a sliced onion. Then grill them at a med-low temp to cook and then finish them at a higher temp to make a nice crust. I've also gone the par-boil route and then just finish them on the grill, but when its hot out it is nice to just avoid that step and do them from start to finish on the grill.

Building on what cm82 suggested above, try tossing them with blue cheese and then put them while still warm on some arugula or watercress. The warmth of the potato wilts the greens and softens the cheese.

one of my favorite camping foods. i cut them bite size with a chopped onion and chopped carrot. butter, s&p, garlic, parsley wrapped up tight in foil then thrown on the grill/fire for 30 minutes.

Foil packs always good. (a little butter or oilive oil, some seasonings or herbs, sweet onions and mushrooms....yum.)

One of my favorite ways to cook potatoes on the grill-- Scrub potatoes well, poke and microwave until almost done but still very firm. While still warm split the potatoes in half, season (salt, pepper, garlic, herbs) and drizzle with olive oil to coat completely. (I do this in a baking pan or cookie sheet for large batches.) (You can use a combo of olive oil and Italian dressing if yo're in a super hurry.) Let rest for at least 10 minutes to allow the flavors to absorb. Then grill until golden and crispy around the edges. Return to the seasoned oil pan and dust with parmesan (optional) and chopped parsley.

Parboil them first-thick slice and dab with oil. Salt and pepper or a more complex mix of spices. Throw on a little blue cheese, olive oil and scallions and it's a meal!

Look up some of Bobby Flay's recipes on www.foodnetwork.com. He has presented some tasty looking recipes.

I usually halve the potatos, make a little hollow to pack with whatever you want, butter, bacon, cheese etc, put the potato back together and wrap in foil.

However I tend to put my potatos under the grill, instead of ontop, an inch or two away from the coals, saves room on the grill for meat and the high heat will get your potatos cooked faster, in time to be served with the main course.
You can turn them a few times during cooking with a stick.

Set up a charcoal grill for indirect cooking. Add wood chips so it's good a smoky. Put russet potatoes directly on the grill grate (no foil as it defeats the purpose of the smoke). Let them roast for an hour or so then enjoy with the usual baked potato toppings. So good.

i like doing sliced red potatoes with oil and parsley in aluminum foil with a squirt of lemon over them at the end. yummers.

You guys are making me hungry....

Levity first - I wouldn't recommend throwing them up on the bbq. ewww

Seriously, keeping them whole, wash and brush some olive oil on them. Poke them with a fork a couple times, wrap them in foil and place them on the coals or place them on the grate if using gas and use the normal oven bake times you'd normally use. OR you could get all fancy schmancy and slice them up, etc... No one way is better than the other, IMHO. Some just require more effort than others. They're all good.