Preheat your barbecue until it reaches 375ºF in the chamber. While that occurs, combine the oil, cumin, paprika, oregano, salt and pepper in a bowl large enough to hold the chicken. Add the chicken and toss to coat. Set the chicken on one side of the barbecue, setting the thigh pieces skin-side up. Turn the heat off under the chicken. Leave the other side of the barbecue turned on. Close the lid and cook for 30 to 40 minutes, or until the temperature of the chicken registers 170ºF (77ºC) when an instant-read meat thermometer is inserted deep into one of the thighs. (As the chicken cooks, check the temperature in the chamber and ensure it stays around 375ºF, adjusting the flame as needed to achieve that.)
Move the chicken to the lit side of the barbecue. Quickly grill on both sides until lightly charred and nicely coloured. Now move the chicken back to the unlit side of the barbecue, setting the thigh pieces skin side up. Brush the thighs and drumsticks with the barbecue sauce, let the sauce heat through a few minutes, and the chicken is ready to serve.