In a large skillet over medium heat, add the ground beef, diced onion, and sliced mushrooms. Cook until the beef is browned and the mushrooms are tender, about 7-10 minutes. Drain any excess fat.
Stir in the minced garlic, oregano, salt, and black pepper. Cook for an additional 2 minutes until the garlic is fragrant.
Remove from heat and mix in the shredded cheddar cheese until melted and well combined.
While the beef mixture cools slightly, prepare the tzatziki sauce. In a bowl, combine the Greek yogurt, grated cucumber, dill, lemon juice, and garlic powder. Mix well and refrigerate for at least 10 minutes to allow the flavors to meld.
To assemble the pita pockets, gently open each pita and fill with the cheesy beef and mushroom mixture.
Drizzle with tzatziki sauce before serving.