Whisk the olive oil, lemon juice, red wine vinegar, garlic, oregano, salt and pepper in a 2 L / 2-quart bowl. Add the chicken cubes and toss until every piece is coated. Cover and refrigerate for 30 minutes, and no longer than 2 hours.
While the chicken marinates, squeeze the grated cucumber in a clean kitchen towel until it stops dripping. Stir it into the Greek yogurt with the grated garlic, lemon juice, dill and salt, then chill. Dice the salad cucumber, tomatoes and red onion, halve the olives, and keep it all cold and undressed.
Lift the chicken from the marinade, let the excess drip off, and pat the cubes dry with paper towels. Heat a 30 cm / 12-inch skillet with 1/2 tsp olive oil over medium-high heat until it shimmers, about 2 minutes. Add half the chicken in a single layer with space between the cubes. Cook 7 to 8 minutes, turning every 2 minutes, until browned on all sides, opaque white in the center and 165 F / 74 C on an instant-read thermometer.
Move the first batch to a plate, wipe the skillet, and repeat with the remaining 1/2 tsp oil and chicken. Tent all the chicken with foil and rest it 3 minutes. Meanwhile, warm the pitas in the same dry skillet over medium heat, 30 seconds per side.
Toss the chopped vegetables and olives with the 2 tsp red wine vinegar and top with the feta. Divide the chicken, tzatziki, salad and pitas among four plates and add a lemon wedge to each.