Try this recipe for Zesty BBQ Drumsticks and have a Yummer Summer! In a gallon-size resealable plastic bag, combine olive oil, BBQ seasoning rub, lime juice and zest, garlic, salt and pepper. Add drumsticks and toss to coat. Marinate in the refrigerator for 15 to 30 minutes or up to 8 hours. Preheat broiler or grill to medium-high heat. Grill drumsticks 14 to 16 minutes, turning occasionally, until chicken is cooked through and golden brown.