In a small bowl, mix together 1 tablespoon garlic powder, 1 tablespoon dried oregano, 1 tablespoon paprika, 1 teaspoon dried rosemary, and 1 tablespoon packed brown sugar; season with salt and freshly ground black pepper to taste (about 1 1/2 tsp salt and 1 tsp pepper suggested).
Pat the quartered chicken dry. Rub the herb-sugar mixture generously all over the chicken pieces. Let sit at room temperature for up to 1 hour.
Prepare the grill for medium-high heat. If using wood chips, sprinkle them on hot coals or place drained chips in a perforated foil packet and position on a gas grill.
Oil the grill grates with about 2 tablespoons oil to prevent sticking.
Place the chicken pieces directly over the medium-high heat and grill, turning once, until well browned, about 3 to 5 minutes per side.
Move the chicken to the cooler (unheated) side of the grill and close the lid; cook 5 minutes.
Brush both sides of each piece liberally with barbecue sauce and continue grilling with the lid closed for an additional 5 minutes.
Continue cooking, turning as needed, until the internal temperature of the chicken reaches 165°F and the sauce is caramelized on the surface. As pieces finish (breasts will finish first), transfer to a platter and tent loosely with foil until serving.