Start by preparing the marinade. In a bowl, whisk lemon juice, olive oil, oregano, salt, and black pepper. Use your hands to mix really well and place the chicken and marinate for about 2 hours.
In the meantime prepare the tzatziki sauce: Place cubed cucumbers in strainer set over bowl and mix with garlic, yogurt, and dill. Season to taste with salt. Refrigerate until ready to serve.
Cut the chicken breasts into 2 cm pieces . Heat grill or grill pan over medium-high heat. Place chicken skewers on grill (or cook in batches on grill pan) until well browned turning evenly to cook on all sides, about 5 minutes total. (Adjust temperature of grill if necessary). Remove chicken to a serving platter and let rest 3 minutes. Meanwhile, briefly grill pitas and keep warm.
To serve the chicken souvlaki (skewers), slice 1 red onion, 2 tomatoes and place on a platter. Put a thick line of tzatziki sauce down the center of the pita. Top with the cooked chicken, sliced onion and tomatoes. Fold the pita up like a taco to eat.