Red bell peppers are a good source of lycopene, an antioxidant pigment that may help ward off stroke and some types of cancer.
Combine 6 Tbsp. vinegar, 2 Tbsp. oil, garlic, oregano, and 1/2 tsp. each salt and black pepper in a small bowl. Divide marinade between 2 ziplock backs. Place chicken in 1 bag and bell peppers in other, turning bags to coat. Marinate at room temperature, turning bags occasionally, for 30 minutes.
Preheat grill or a lightly oiled grill pan to medium-high heat. Remove chicken and bell peppers from bags; discard marinade. Grill chicken and bell peppers, sprinkling with salt and black pepper and turning, until chicken is cooked through and bell peppers have softened, 5 to 6 minutes. Transfer to a cutting board; let chicken rest for 5 minutes before slicing.
Slice chicken and bell peppers into strips; arrange over romaine. Top with cheese. Drizzle with remaining 2 Tbsp. vinegar and 1 Tbsp. oil. Sprinkle with black pepper and serve.