Heat oil in a skillet over medium-low heat. Add garlic, cumin, salt, pepper, oregano, and achiote paste (See Note) and cook for several minutes, stirring. Remove from heat and allow to cool. Set aside.
Add cooled achiote garlic mixture, orange juice and lime juice to a bowl and whisk to combine. Reserve 1/4 cup of marinade in the bowl to use for basting the chicken as it grills. Add remaining marinade to a 1 or 2 gallon Ziploc freezer bag. Add chicken pieces, seal bag, and massage the bag, then transfer to a refrigerator to allow the chicken to marinate a minimum of 45 minutes, but I recommend 4 hours or up to overnight (8 hours).
Grilling: Preheat grill to 350°F to 400°F. Remove chicken from marinade, then discard the bag of marinade. We will be grilling with indirect heat.
For indirect cooking on a charcoal grill, place your lit charcoal across half of the grill and leave the other half empty. To indirect cook on a gas grill, only turn half of the burners on (for mine I keep the top Medium, middle Low and the bottom Medium).
Grill chicken, covered, until an instant-read thermometer registers 165°F when inserted into thickest portion of each piece, about 20 to 30 minutes. As chicken is cooking, use reserved marinade to baste the pieces every few minutes. I do not turn the chicken pieces over during cooking, as there is no need for grill marks.
Cook on the Stovetop: Heat 2 tablespoons of oil in a large skillet over medium-high heat. Add the chicken pieces and cook for 5 to 10 minutes per side, depending on the thickness of the meat, until cooked through and the internal temperature reaches 160-165°F on an instant read thermometer.
Bake in the Oven: Preheat the oven to 425°F. Arrange the chicken pieces in a single layer on a large foil-lined baking dish. Bake the chicken until the internal temperature reaches 160-165°F on an instant-read thermometer (chicken thighs and tenders need 15 minutes. Medium to large breasts need 20 to 25 minutes).
Most achiote paste is packaged in 4 ounce cubes. We will only need 2 ounces, save remainder for another use. I break up and smash the achiote paste as best I can in the marinade mixture, making sure it's blended in. You can also puree in a blender for a smoother mixture after it cools.BELOW is an achiote spice blend substitute, enough for this recipe. Achiote Paste Alternative
3 tbsp paprika
1 tbsp white vinegar
2 tsp dried Mexican oregano
4 garlic cloves, minced
1/2 tsp ground cumin
Combine in a small bowl and use to marinate chicken along with other main recipe seasonings and usually citrus juice.Marinate a minimum of 45 minutes, but I recommend 4 hours or up to overnight. Unlike beef, poultry doesn't have any tough connective tissue that needs to be tenderized. It starts to break down and the texture gets mushy. While it’s OK to marinate overnight, any longer would be far too long if the marinade or rub has a high acid base (vinegar, wine, citrus juice and tomatoes, etc). So when using a highly acidic marinade for chicken, add a little olive oil and/or minimize marinating time.