Lightly sprinkle work surface with cornmeal or the flour. Place dough directly in the middle of the work surface. Roll out gently into either a 12-inch rectangle or circle 1/4-inch thick. Brush both sides with olive oil and set aside until ready to grill.
Gently place dough in center of the cooking grate, directly over the heat for 2-4 minutes, until the bottom of the crust is well marked and browned. Remove from grill and sprinkle the tomatoes, then pepperoni and mushrooms., on the cooked side of crust. Top with mozzarella and dried oregano. If using a gas grill, switch to indirect heat. Return pizza to the center of the cooking grate and cook with the lid closed until bottom is browned, toppings are warm and cheese is bubbly, about 5-10 minutes.
Remove from grill and season with salt and pepper to taste. Slice and serve immediately.