Combine the orange juice, lime juice, salt, pepper, oregano, cumin and garlic in a shallow baking dish. Marinate the chicken in this mixture for at least 30 minutes.
Preheat oven to 400 degrees. Bake the chicken for 30 minutes, until tender. Shred with forks.
Increase the oven temperature to 450 degrees. Heat tortillas in the microwave for 30 seconds. Keep tortillas covered with damp paper towels while you're working. While still warm, brush both sides with oil. Place a thin line of shredded chicken and cheese in the center of each tortilla. Roll and place, seam side down, onto a baking sheet.
Bake taquitos 12-14 minutes, until golden. Flip and bake 8-10 more minutes. Serve warm.