Place strainer over large bowl. Line strainer with 3 layers of cheesecloth. Spoon yogurt into cheesecloth-lined strainer; let stand at room temperature to drain 3 hours (liquid will drain out and yogurt will thicken). Transfer yogurt to medium bowl; discard liquid.
Meanwhile, coarsely grate cucumber. Place in another strainer; let stand at room temperature until most of liquid drains out, about 3 hours. Discard liquid. Squeeze excess moisture from cucumber.
Mix cucumber, dill and garlic into yogurt. (Can be made 1 day ahead. Cover and refrigerate.)
Mix tomato into yogurt. Season to taste with salt and pepper and serve.
