In a large skillet over medium heat, add the ground beef, onions, and garlic. Cook until the beef is browned and the onions are translucent, about 5-7 minutes. Drain excess fat if necessary.
Add the diced mushrooms, oregano, salt, and black pepper to the skillet. Cook for an additional 5 minutes until the mushrooms are tender. Stir in the mozzarella cheese until melted and well combined. Remove from heat.
In a separate bowl, mix the Greek yogurt, grated cucumber, lemon juice, dill, and garlic powder to create the tzatziki sauce. Stir until smooth and well combined.
Warm the pita pockets in a toaster or on a skillet for about 1-2 minutes until soft. Carefully open each pita pocket and fill with the cheesy beef and mushroom mixture.
Serve the pita pockets warm with a generous drizzle of tzatziki sauce on top.