Place chicken in a plastic bag set in a shallow dish. Pour Italian dressing over chicken. Close bag. Marinate in the refrigerator for at least 4 hours or up to 24 hours, turning the bag occasionally.
Drain chicken, discarding marinade. Place chicken on the unheated rack of a broiler pan. Broil 4 to 5 inches from the heat for 12 to 15 minutes or until no longer pink and juices run clear, turning once.
For sauce, in a medium saucepan, cook mushrooms and garlic in butter or margarine until mushrooms are tender, stirring occasionally. Stir in flour and pepper. Stir in milk all at once. Cook and stir until thickened and bubbly. Cook and stir 1 minute more. Add cheese and stir into the sauce until cheese is melted.
To serve, spoon hot cooked rice onto warm serving plates. Arrange a chicken breast half on each. Ladle sauce over all. Makes 4 servings.