Make the tzatziki: Put the Greek yogurt in a cheesecloth lined strainer set over a bowl and let it drain for 24 hours in the refrigerator. Put the strained Greek yogurt in a small bowl and mix in the rest of the tzatziki ingredients. Cover the sauce and put it in the refrigerator to chill, for at least an hour.
Make the spice mix: Mix together the spice mix ingredients in a small bowl.
Grill the mushrooms: Preheat an outdoor grill to medium-high heat. Brush one of the portobello mushroom caps liberally with olive oil, sprinkle the cap with the spice mix, then rub it gently into the mushroom. Repeat with the remaining mushrooms. Grill the mushrooms for 3 to 4 minutes on each side, starting with the cap side first, until the mushrooms are dark and tender. Remove the mushrooms from the grill and cut them into thin slices.
Assemble the pita sandwiches: Wrap the pitas in a damp paper towel (to prevent them from drying out) and warm in the microwave or oven. Spread a spoonful of the tzatziki sauce down the center of one of the pitas, add a few pieces of lettuce, then add grilled mushroom slices, sliced tomato, and sliced onion, then top it with another generous spoonful of tzatziki. Fold the pita over and wrap with a piece of foil to hold the sandwich together for easy eating. Repeat with the other pitas. Serve immediately.
