Combine cherry tomatoes and cucumbers in a salad bowl. Sprinkle generously with salt, stir well, then leave to sit while you prepare the purslane.
Rinse purslane well, then pat dry with a tea towel. Pick off any large leaves and the tender stems at the top and place in a bowl. Discard the larger sections of the stems.
Check tomato-cucumber mixture; by now, tomatoes should have released juices thanks to the salt. If not, stir well and let sit 5 minutes more.
Add purslane to the salad bowl with the tomatoes and cucumbers. Drizzle with olive oil and toss well. Serve immediately.