In a medium bowl, combine smoked paprika, ground cumin, dried oregano, garlic powder, onion powder, black pepper, and salt.
Add 1 tablespoon olive oil to the spice mixture and stir to create a marinade.
Add the tempeh strips to the marinade and toss to coat evenly. Let marinate for 10 minutes.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat.
Add the marinated tempeh strips to the skillet and cook for 5-7 minutes, turning occasionally, until golden brown and slightly crispy.
While the tempeh cooks, prepare the vegan tzatziki sauce by combining vegan yogurt, grated cucumber, lemon juice, fresh dill, and minced garlic in a small bowl. Mix well and refrigerate until ready to use.
Warm the pita breads in a dry skillet or microwave for about 30 seconds until soft and pliable.
To assemble each gyro, spread 2 tablespoons of vegan tzatziki sauce onto the center of a warm pita bread.
Layer with cooked tempeh strips, sliced red onion, tomato slices, shredded lettuce, and Kalamata olives.
Fold the pita over the fillings and serve immediately.