Puree garlic, soy sauce, lime juice, oil, cumin, paprika, oregano, and a pinch of black pepper in a blender.
Place chicken in a large ziplock bag and add marinade. Refrigerate to marinate for 8 to 24 hours.
In a blender, combine all ingredients except the olive oil and process until smooth. With the motor running, gradually add olive oil. Transfer to a bowl and refrigerate until ready to use.
Preheat grill to medium-high heat (about 350 degrees). Remove chicken from marinade and shake off any excess. Place on grill grates. Cook for 5 to 6 minutes with the grill cover closed. Use tongs to flip the chicken thighs and cook for an additional 5 to 6 minutes. Check that internal temperature reaches 165°F.
Alternatively, preheat the oven to 500 degrees. Arrange chicken pieces in a 13x9 inch roasting pan and add 1 cup water. Bake uncovered for 30 minutes, then tent with foil and continue cooking for another 15 minutes, until chicken is fully cooked (internal temperature should reach 165°F).