To prepare chicken marinade, whisk all marinade ingredients except chicken breasts together in a bowl. Combine marinade with chicken breasts in a large zip-lock bag, seal and gently shake to evenly distribute marinade over chicken. Place in refrigerator and chill for 2 to 4 hours.
Meanwhile, prepare the cucumber relish. Lay cucumber slices in a sieve placed over a bowl and sprinkle with pinch of salt. Leave for about 30 to 40 minutes, or until cucumbers soften and their liquid begins to drain into the bowl. Stir occasionally.
Combine the tomato, onion, olives, parsley, dill, garlic, olive oil and lemon juice in a medium bowl. Stir in drained cucumbers, then cover the bowl and refrigerate for 1 to 2 hours.
Grease grill and preheat to medium-high. Remove chicken from zip-lock bag and discard marinade. Grill chicken breasts until cooked through, about 12 minutes. Remove chicken from grill, let rest for a few minutes and slice thinly on the diagonal. Lay chicken slices on top of each pita round and divide relish evenly among them. Garnish with a little extra yogurt and serve.