Season chicken with salt and pepper. In a large (5-quart) saucepan with a tight-fitting lid, heat oil over medium-high heat; swirl to coat bottom of pan. Cook chicken, turning when it easily releases from the pan, until golden, 4 to 6 minutes. Transfer to a plate.
Add mushrooms; cover, and cook until softened, about 5 minutes. Add garlic, tomatoes, and oregano. Reduce heat to medium-low; cook, covered, until tomatoes have broken down, 10 to 15 minutes.
Return chicken and any accumulated juices to pan; cover, and cook until chicken is opaque throughout, 4 to 6 minutes. Turn chicken to coat with sauce, and serve.
Before It's Cooked: To catch any leaks, refrigerate packages of raw chicken on a rimmed baking sheet.