Whisk the olive oil, lemon juice, minced garlic, dried oregano, salt, and black pepper in a large bowl. Add the cubed chicken breast, toss to coat, and marinate for 15 minutes.
Preheat the grill to medium-high heat (about 400°F).
Thread the marinated chicken onto skewers, alternating with chunks of red onion, cherry tomatoes, and green bell peppers.
Grill the kebabs for 10-12 minutes, turning occasionally, until the chicken reaches an internal temperature of 165°F and the vegetables are tender-crisp.
Place the pita bread on the grill for the last 1-2 minutes of cooking to warm them through.
Garnish the cooked kebabs with fresh chopped dill and serve immediately with warm pita bread and tzatziki sauce.