Preheat a grill to high. Rub the turkey parts with olive oil and season with salt (omit if the turkey is brined) and pepper. (Apply a rub or paste, if using) Just before grilling, lower the heat to low and oil the grate. Arrange the breast halves on the oiled grate and cook, covered and turning halfway through, about 30 minutes. Add the leg quarters and wings, cover and grill, turning every 15 minutes. (Baste all parts twice with a sauce, if using, during the last 25 minutes of grilling.) Cook until an instant-read thermometer registers 160 degrees when inserted into the meaty part of the thighs and breasts, about 45 minutes.
Transfer to a work surface and tent loosely with foil for 15 minutes before carving.