Heat olive oil in a small saucepan over medium-high heat until very hot. Meanwhile, mash cumin, salt, oregano, and garlic together in a heat-proof bowl until a paste forms.
Remove oil from the heat and whisk in paste; allow to cool slightly. Stir in lime juice and orange juice.
Place chicken into a large resealable bag; pour in marinade and shake until chicken is evenly coated. Refrigerate for at least 4 hours, or overnight.
Preheat the oven to 375 degrees F (190 degrees C).
Remove chicken from marinade and shake off excess. Discard remaining marinade. Arrange chicken in a roasting pan.
Bake in the preheated oven until no longer pink at the bone and the juices run clear, about 1 hour. An instant-read thermometer inserted near the bone should read at least 165 degrees F (74 degrees C).