To prepare chicken, dissolve achiote seasoning in vinegar and oil. Add chicken, turn to coat well, and marinate in the refrigerator 1 hour.
Prepare the grill. Remove chicken from marinade, reserving marinade. Grill chicken until done, about 7 minutes per side. Remove chicken to platter, and tent with aluminum to keep warm.
To prepare sauce, pour reserved marinade into a saucepan; add garlic and juices. Bring to a boil, reduce heat and cook, stirring, until sauce is the consistency of a thin cream sauce. (If sauce gets too thick, thin with a little water.) Add salt and pepper. Serve sauce over chicken. Garnish with chopped cilantro.