In a small bowl, whisk together garlic, lemon juice, extra-virgin olive oil, oregano, salt, black pepper, cayenne, and paprika.
Place chicken drumsticks and chicken thighs in a large bowl, pour [1/2] of dressing over chicken, mix to coat evenly, and set in the refrigerator to marinate for 1 hour.
Preheat the grill to medium-high heat.
Place chicken evenly on the grill, and cook for 5 minutes per side.
Reduce heat to medium-low, cover the grill, and cook chicken for 15 minutes pre side or until juices run clear and internal temperature of chicken reads 175ºF.
Remove chicken from the grill, and let rest for 5 minutes before serving warm.