Combine the lime juice, 2 tablespoons of the olive oil, oregano, 1/2 teaspoon of the pepper and 1/2 teaspoon of the salt in a glass dish. Mix well, and add the chicken breasts. Cover loosely and marinate, turning occasionally, for 2 hours. Preheat oven to 350F and prepare hot coals for grilling. In a shallow roasting pan , combine the slivered onions and peppers with the chicken breasts and marinade. Cover the pan with aluminum foil, and bake until the chicken is just opaque, 15-20 minutes. Remove the chicken breast to a platter, and set aside. Transfer the onions and peppers to a heavy skillet, and add the remaining 4 tablespoons olive oil, and any pan juices. Sprinkle with the remaining 1/2 teaspoon salt and pepper. Cook over medium heat, stirring often, until the onions and peppers are soft and slightly browned, about 15 minutes. Drain and set aside. Place the chicken breasts over very hot coals and grill until they are cooked through but still moist, about 3 minutes on each side. Slice them into strips . Serve with corn or flour tortillas and a variety of your favorite condiments.