Season the chicken drumsticks on both sides with salt and black pepper.
In a small bowl, combine the olive oil, garlic, thyme, rosemary, and lemon zest. Mix well.
Add the marinade to the chicken drumsticks. Cover with plastic wrap then let marinate in the refrigerator for at least 2 hours to overnight. Alternatively, place the chicken in a Ziploc bag.
Preheat the grill to HIGH heat and oil the grates. Place the marinated chicken on the grill and cook for 5-7 minutes on each side.