In large bowl or baking dish, mix together oil, cumin, oregano, paprika, salt and pepper. Add chicken pieces and turn to coat.
Place chicken pieces -- thighs skin-side up -- on one side of the grill in barbecue preheated to 375°F (190°C). Turn off flame directly under chicken, but keep flame lit on other side of grill. Close lid and cook 30 to 40 min., keeping temperature of barbecue at 375°F (190°C). Use an instant-read thermometer to check that it registers 165°F (74°C) temperature when inserted into thickest section of chicken thighs.
Move chicken pieces to lit side of barbecue 2 min. to lightly char skin. Move back to unlit side of barbecue. Brush with barbecue sauce; cook another 2 min.
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