Preheat a gas grill to medium-high or start the coals on a charcoal grill.
While the grill is heating, mix together the rub by combining the brown sugar, chili powder, cumin, salt, pepper, and olive oil in small bowl.
Using your hands, coat each chicken leg generously with the rub. Place the seasoned chicken legs on the grill, with the meaty end closest to the heat source.
Cook on medium-high heat for 5 minutes per side. Brush the chicken legs with barbecue sauce each time you turn them. Turn the legs 3 to 4 times, for a total cooking time of about 18-20 minutes. The chicken is done when it reaches an internal temperature of 165 degrees F.
Remove the chicken from the grill and let rest for 5 minutes before serving with additional bbq sauce for dipping.
Notes
If you don't want to make homemade BBQ sauce, you can use your favorite store-bought brand. Want to give your chicken legs a kick? Try this Smokey Chipotle Barbecue Sauce instead.