Place 8-10 wooden skewers in a shallow pan of water to let them soak before grilling. It's best if they soak at least 30 minutes.
Cut the chicken breasts into 1-inch pieces and place them in a large bowl. Add 3 tablespoons of fresh lemon juice from one lemon. Add the olive oil, garlic, oregano, 1 1/2 teaspoons salt, and 1/2 teaspoon ground black pepper. Toss and let the chicken marinade for 20-30 minutes.
Meanwhile, in a second bowl mix the ingredients for the Tzatziki Sauce. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Stir well, then place in the refrigerator until ready to serve.
Preheat the grill (or a grill pan) to medium heat. Slice the remaining lemons into thin rounds. Slide 7-8 pieces of chicken onto each skewer, with a folded lemon slice in between every 2 or 3 chunks. Once all the skewers are ready, grill the Chicken Souvlaki for 5 minutes per side, 10 minutes total. Be careful that the wooden skewers are not resting directly on the grill, or they will burn up.
Break open one large chicken chunk to make sure they are cooked through. Then serve the Chicken Souvlaki with Tzatziki Sauce for dipping.