Place the chicken in a medium bowl or sealable bag. Add the orange juice, lime
garlic, oregano, cumin, salt, pepper, and cayenne. Seal and store in the refrigerator for up to 8 hours or leave at room temperature for up to 30 minutes if cooking immediately.
Prepare a charcoal fire using about 6 pounds of charcoal and burn until the coals are completely covered with a thin coating of light gray ash, 20 to 30 minutes. Spread the coals evenly over the grill bottom, position the grill rack above the coals, and heat until medium-hot (when you can hold your hand 5 inches above the grill surface for no longer than 3 or 4 seconds). Or, for a gas grill, turn on all burners to high, close the lid, and heat until very hot, 10 to 15 minutes.
Apply some oil to the grill grate. Place the chicken on the grill, leaving plenty of space between each piece. Grill until seared, 3 to 4 minutes for breasts.
Move the chicken pieces to medium-low heat or reduce the heat to medium; continue to grill, turning occasionally and squirting with the marinade, until the juices run clear when pierced and an instant-read thermometer registers 165 degrees, about 35 minutes total.