Cover one side of a grill with a large sheet of foil; fold up the edges to make a rimmed tray. Preheat the grill to medium high. Rub the chicken all over with the garlic powder and oregano; season generously with salt and pepper. Toss the green beans, tomatoes, garlic and olive oil in a large bowl; season with 1/2 teaspoon salt and a few grinds of pepper.
Brush the grill grates with olive oil on the side without the foil tray. Place the chicken on the grates and grill, turning halfway through, until marked and cooked through, about 15 minutes. Remove to a plate. Meanwhile, spread the vegetable mixture on the foil tray and grill, stirring occasionally, until tender and charred in spots, 10 to 12 minutes. Transfer the vegetables to a bowl; add the vinegar and dill. Season with salt and pepper and toss.
Divide the chicken and vegetables among plates. Top the vegetables with the cheese. Drizzle with more olive oil.