In a large mixing bowl, mix together lime juice, vinegar, salt and sugar.
Combine olive oil, cumin, paprika, and chile powder in a small saucepan; heat over medium heat until aromatic and oil begins to bubble, about 3 minutes. Add hot oil mixture to lime juice mixture. Let cool. Add chicken pieces to bowl and coat well with marinade. Cover and refrigerate for at least 4 hours or preferably overnight. Let the chicken come to room temperature before cooking it, about 30 minutes.
Build a medium fire in a grill and add chunks of a flavorful wood such as maple, oak or hickory. When the wood has burned down to embers, place the chicken, skin side down, directly over the coals. Cook for 2 minutes, then rotate the grill grate so the chicken is away from the fire. Cover the grill and cook until an instant-read thermometer registers 160 degrees F when inserted along the leg about, about 20 minutes.