Dredge chicken lightly in flour. Heat oil over medium-high heat. Fry chicken pieces briefly, turning to brown evenly. Add onion and celery; cook 1 to 2 minutes. Stir in wine and chopped tomatoes. Season with oregano, and salt and ground pepper to taste. Cover, reduce heat, and simmer gently for 30 minutes, turning pieces once.
Pour in balsamic vinegar and mushrooms, and cook another 5 to 10 minutes.
