Peel cucumbers leaving some green skin in alternating strips; slice as thinly as possible. Slice onions as thinly as possible. In a colander, toss cucumber and onion slices with 2 teaspoons salt. Weight with a plate and let drain 20 minutes. Quarter, then thinly slice tomatoes.
In a large bowl of cold water soak staled bread just until it loses its firmness and becomes spongy, 30 seconds to 1 minute. (If you have let your bread get stale for less than 2 days, just run it briefly under cold, running water.) Using your hands, gently squeeze out water without compacting bread too much. Tear into pieces about 1/2-inch wide and 3-inches long.
In a large bowl combine soaked bread, cucumbers and onions, tomatoes, basil, oil and vinegar. Season with salt and pepper. Serve immediately.