In a large mixing bowl or resealable bag, combine the orange juice, lime juice, olive oil, minced garlic, achiote powder, cumin, coriander, oregano, salt, and black pepper. Whisk or shake until well combined.

Add your choice of chicken thighs, legs, or breasts into the marinade, making sure each piece is well coated. Seal the bag or cover the bowl with plastic wrap, and allow the chicken to marinate for at least 30 minutes at room temperature, or ideally 4 hours in the refrigerator.

When you're ready to cook, preheat your grill to medium-high heat, or preheat oven to 400°F (204°C) and prepare a baking pan with aluminum foil.

Remove the marinated chicken from the bag or bowl, and place on the grill skin-side down for about 20 minutes, flipping halfway through, until the internal temperature reaches 160°F-165°F.

For oven preparation, arrange the marinated chicken skin-side up on a wire rack in the baking pan. Bake for 35-45 minutes, flipping midway, until the chicken reaches an internal temperature of 165°F.

Once cooked through, remove the chicken from the grill or oven and let it rest for about 10 minutes before serving with lime wedges and chopped cilantro.