Preheat a grill to high. In an 18-inch roasting pan, add the potatoes, sausage and 4 cups water. Place on the grill, close the grill lid and bring to a boil. Add the clams; scatter the lemon rounds, garlic and thyme on top. Cover with foil or the roaster lid, then close the grill lid and cook for 15 minutes. Stir well, add the shrimp and corn, cover, close the grill lid and cook until the clams open, 10 to 15 minutes.
Transfer the shellfish, sausage and vegetables to bowls or platters and serve with the lemon wedges, melted butter and Old Bay Seasoning. Ladle the clam broth into large bowls and serve with the crusty bread.