Preheat your grill to medium-high heat.
In a large bowl, combine the olive oil, paprika, cumin, garlic powder, onion powder, salt, black pepper, cayenne pepper (if using), and lime juice. Mix well to create a marinade.
Add the chicken drumsticks to the marinade and toss until they are evenly coated. Let them marinate for at least 30 minutes, or up to 2 hours in the refrigerator.
While the chicken is marinating, prepare the Zesty Cilantro Jalapeño Sauce by blending cilantro, jalapeño, Greek yogurt, lime juice, minced garlic, salt, and pepper until smooth.
Once the grill is hot, place the marinated drumsticks on the grill. Cook for about 25-30 minutes, turning occasionally, until the chicken is cooked through.
Remove the drumsticks from the grill and let them rest for 5 minutes before serving.
Serve the drumsticks with the Zesty Cilantro Jalapeño Sauce on the side for dipping.