Prepare the beets: Preheat oven to 400 degrees F. Rub beets with olive oil and season with cumin, salt, and pepper. Wrap each beet in aluminum foil and place on a baking sheet. Roast for 60 minutes or until fully cooked through (when a sharp knife pierces the beet easily). Allow to cool completely. Once cooled, peel the beets, dice into 1-inch chunks, and set aside.
After the beets have been roasting for 40 minutes, in a medium bowl, toss the chickpeas with 1 tablespoon olive oil, cumin, garlic powder, salt, and pepper. Transfer to a separate baking sheet and roast for 12 minutes. Then, stir, and roast for 8 to 10 minutes more, until tender and dark golden. Set aside.
In a large bowl, combine tomato, cucumber, parsley, scallion, radishes, and tzatziki. Spoon into pitas along with roasted beets and chickpeas. Serve immediately.