In a small bowl, mix the balsamic vinaigrette, juice from 1/2 lemon, oregano and 1/2 teaspoon black pepper. Place chicken in a large resealable plastic bag. Pour marinade over the chicken, seal, and refrigerate for at least 1 hour.
Preheat an outdoor BBQ for high heat.
Toss the grated cucumber with 1 teaspoon sea salt, and allow to sit at least 5 minutes. In a medium bowl, mix the yoghurt, soured cream, 1 tablespoon lemon juice, rice vinegar and olive oil. Season with garlic, fresh dill and dried herbs. Squeeze the cucumber to remove any excess water; stir into sauce. Season to taste with salt and pepper. Refrigerate until ready to use.
Remove chicken from marinade and place on prepared BBQ. Discard remaining marinade. Cook chicken until juices run clear, about 8 minutes on each side. Remove chicken from heat, and allow to sit about 10 minutes before slicing into thin strips.
Place pitta rounds on the BBQ, and cook for about 2 minutes, until warm, turning frequently to avoid burning. Arrange warmed pitta, sliced chicken, lettuce, onion, tomato, olives and hot peppers on a serving platter. Serve tzatziki and feta cheese in separate bowls on the side. Stuff pitta pockets with chicken and toppings to serve.