In a medium bowl, whisk together the lemon juice, red wine vinegar, olive oil, oregano, salt, and black pepper.
Place the chicken pieces in another bowl and mix with the grated garlic. Add 7 tablespoons of the marinade and toss until the chicken is evenly coated. Cover and refrigerate for about 2 hours, tossing occasionally. Refrigerate the remaining unused marinade separately.
While the chicken marinates, prepare the tzatziki. Place the diced cucumber in a strainer over a bowl. Toss with 1/2 teaspoon salt and let it drain for about 30 minutes.
Gently pat the cucumber dry with paper towels. Transfer it to a small bowl and stir in the grated garlic, Greek yogurt, and dill. Season with salt to taste, cover, and refrigerate.
Thread the marinated chicken onto 8 to 12 skewers. Discard the marinade that has been in contact with the raw chicken.
Heat a grill or grill pan over medium-high heat. Grill the chicken skewers, turning them regularly, until browned and cooked through, about 5 minutes depending on the size of the pieces. For food safety, chicken should reach 165°F (74°C) in the thickest piece.
Transfer the chicken to a serving platter and let it rest briefly. Meanwhile, warm the pita breads on the grill.
For the Greek salad, whisk the 2 tablespoons olive oil into the reserved unused marinade. Add the tomatoes, red onions, cucumber, Kalamata olives, feta, and parsley. Toss gently and season with salt and black pepper.
Serve the chicken souvlaki with warm pita, tzatziki sauce, and Greek salad. Remove the chicken from the skewers and tuck it into the pita with plenty of tzatziki and salad.