Cut chicken breasts crosswise into 2.5 cm (1 inch) strips and cut each strip into 3.5 cm (1 ½ inch) chunks. Place in bowl. Add oregano, garlic, lemon rind and juice and pepper and stir to coat well. Cover and refrigerate for up to 30 minutes.
Meanwhile, in bowl, stir together yogurt, cucumber, garlic, dill and lemon rind until combined. Cover and refrigerate for up to 24 hours.
Preheat broiler to high or if using grill, preheat grill to medium high.
Skewer chicken onto 4 metal or soaked wooden skewers. If broiling, place on foil lined baking sheet and place sheet in oven, about 15 cm (6 inches) from broiler. Broil, turning once, for about 8 minutes or until golden brown and no longer pink inside.
If grilling, place skewers on greased grill over medium heat for about 10 minutes. Turn once.
Serve with tzatziki.
