Place yogurt in a fine sieve set over a small bowl. Let drain for 30 minutes. Combine tomatoes and cucumber in a medium bowl, and sprinkle with 1/2 teaspoon salt. Toss to coat, and let stand for 30 minutes.
Meanwhile, preheat oven to 450 degrees. Sprinkle 1 teaspoon salt on both sides of chicken; season with pepper, and transfer to a rimmed baking sheet. Bake until chicken registers 160 degrees on an instant-read thermometer, 20 to 25 minutes. Let stand until cool enough to handle. Shred chicken into 1-inch chunks, and let cool completely.
Drain tomato-cucumber mixture. Add drained yogurt, chicken, lemon juice, garlic, and remaining 1/2 teaspoon salt. Season with pepper, and stir.
Grill or toast pitas, fill with romaine, and divide chicken salad evenly among all halves. Top with herbs.