In a shallow dish, combine the flour with 1 teaspoon salt and 1 teaspoon pepper. Sprinkle the chicken pieces with 1 teaspoon each salt and pepper. Working in batches, dredge the chicken pieces in the flour mixture, turn to coat and shake off the excess flour.
Heat the oil in a large heavy-bottomed saute pan or Dutch-oven over medium heat. Start browning the chicken in the pan a few pieces at a time to avoid overcrowding. Cook the chicken 5 minutes on each side. Remove and transfer the chicken to a plate.
In the same pan, add the mushroom, onion, red bell pepper, and garlic and saute just until tender. Add the wine and simmer until reduced by half. Add the broth, tomatoes, and oregano. Return the chicken pieces to the pan and turn them to coat in the sauce. Add the thyme to the sauce and bring the mixture to a simmer. Reduce the heat to medium-low, and cover. Continue simmering the mixture for 40 minutes, or until tender.
Serve the chicken cacciatore with crusty Italian bread, over pasta, or over polenta and garnish with fresh basil.
Veal Cacciatore: substitute veal for the chicken pieces and replace the red wine with white wine.
Cook's Note: There are many variations of this dish based on the regions of Italy. In Southern Italy they use red wine and in Northern Italy they use white. Try both to determine your preference.