Preheat grill to 500 degrees. To get a smoky flavor, soak applewood chips in water and put in a grill smoker box and place on the grill.
Use a brined turkey for this method. Remove turkey from brine and dry. Place onion, orange, rosemary, and bay leaves in turkey. Break up garlic and add to turkey.
Cover turkey with dry rub seasoning mix or season generously with salt and pepper. Place turkey on a roasting rack and put rack onto a disposable aluminum roasting pan.
Turn off middle burner of grill and place roasting pan on grate. Lower temperature of front and back burners to maintain a 350-degree grill. Try to open grill lid as little as possible in order to maintain temperature. Cover the breast and legs with aluminum foil if they begin to get too dark.
Grill until temperature between turkey's leg and thigh reads 175 degrees. (Grilling a turkey takes approximately 12 minutes per pound.)Remove turkey from the grill and let rest for at least 30 minutes before carving.