Slice chicken breasts lengthwise into ½ inch (1 cm) strips and place in a resealable plastic bag. Combine oregano, dill, lemon juice, water and vegetable oil and pour over chicken; close bag and turn it until all the strips are coated with the marinade. Let stand at room temperature for no longer than 30 minutes.
In the meantime, combine sauce ingredients and refrigerate until needed.
Preheat barbecue on medium heat. Thread chicken strips onto eight metal skewers; discard marinade. Grill skewers, with the lid down, for 5 minutes per side.
Cut onion into ½ inch (1 cm) slices and grill for a few minutes on each side. Remove from heat and cut slices into four.
Wrap pita breads in foil and warm on the grill. Arrange warm pitas on plates.
Spread yogurt sauce over pitas, top with tomatoes, cucumber, and grilled onions and two skewers of chicken. Fold the sides of the bread up, pull the skewers out of the chicken and eat the pita like a taco.