Place the chicken in a medium bowl and add all of the marinade ingredients. Mix it well, cover with plastic wrap, and marinate for at least 2 hours, preferably overnight.
Grate the cucumber and squeeze the excess liquid out with your hands. Discard the liquid. Place the cucumber in a mixing bowl with the rest of the tzatziki ingredients and mix well.
Heat a large, non-stick skillet over medium-high heat. Add the chicken slices and cook for about 4-5 minutes on each side, until cooked through. Slice the chicken into thin strips.
Warm the pita bread. Add a bit of shredded lettuce, tomato, cucumber, onion, and chicken on top. Drizzle with a generous amount of tzatziki and feta cheese, if desired. Fold the gyro* in half and serve.