Gas grill: Preheat your grill to medium-high heat (about 400°F). Add the chicken drumsticks to clean, well-oiled grates, and cook. Turn each piece after a few minutes, until the skin is browned and crispy on all sides.
Lower the heat to medium low (about 300°F), and close the grill. Cook for about 25 minutes, stopping halfway through to turn the chicken once.
Charcoal grill: Create two heat zones (one with medium high heat and one with medium-low heat) by moving most of the charcoal to one side of the grill.
Begin by placing the pieces of chicken directly over the heat source, turning every few minutes as the skin browns.
Once the skin is browned and crispy, move the chicken away from the heat source, to the indirect (medium-low) heat zone. Cover and cook for about 25 more minutes, stopping halfway through to turn the chicken.