Prepare the tempeh: In a bowl or container big enough to marinate the tempeh, stir together the garlic, white wine vinegar, lemon juice, olive oil, soy sauce, greek yogurt, oregano and salt. Add the sliced tempeh, making sure to coat each slice in the marinade and let sit in the fridge while you prepare the rest of the recipe. You can also do this step the night before.
Preheat oven to 400F. Place the prepared potatoes on a baking sheet. Drizzle with oil and toss to coat. Sprinkle with salt and oregano. Roast for 30 minutes or until golden and crispy. For crispy fries, make sure to spread them on the baking sheet with room between each.
Meanwhile, prepare the tzatziki: Shred the cucumber and squeeze out the liquid using a clean dish towel. Place the cucumber in a bowl, along with the yogurt, lemon juice, olive oil and salt. Stir together until combined.
Cook the tempeh: Heat a non-stick skillet over medium heat. Place the tempeh in a single layer in the skillet, along with some of the marinade. Cook 5 minutes on each side. You can drizzle any remaining marinade on the tempeh during cooking.
Assemble the gyros: Spread the tzatziki on each pita. Add the tempeh, fries and other desired fillings. Serve immediately.