Combine 2 tbsp of olive oil, 1 tbs lemon juice, oregano, and half the garlic in a shallow dish. Season with salt and pepper. Add steak and set aside for 15-20 minutes to marinate.
Meanwhile, toss onion with remaining lemon juice in a small bowl. Season and set aside.
To make the tzatziki, use a teaspoon to scoop out cucumber seeds. Grate the cucumber and squeeze to remove excess liquid. Combine yoghurt, dill, remaining oil and garlic, and 1/4 tsp salt in a bowl. Stir in cucumber.
Heat a barbecue grill on high heat. Reduce heat to medium-high. Add steak and cook for 3-5 minutes on each side for medium-rare or until done to your liking. Transfer to a plate, cover loosely with foil and rest for 5 minutes. Thinly slice.
Drain onion. Warm pita bread on a barbecue for 30 seconds on each side. Spread with tzatziki, and top with lettuce, tomato, steak and onion. Tie with string to form a loose wrap and serve.