Carefully brush vegetable oil over the grill rack, or spray it with cooking spray. Place the chicken on the grill rack over medium heat. Cover the grill; cook 15 to 20 minutes, turning chicken over once with tongs about halfway through cooking. Brush chicken with the barbecue or teriyaki sauce during the last 10 minutes of cooking. Cook until juice of chicken is clear when you cut into the center of the thickest part and an instant-read meat thermometer inserted into the thickest part reads 170°F.