Combine first 4 ingredients in a large bowl, and toss gently. Combine vinegar, oil, salt, pepper, and garlic in a small bowl, and stir well with a whisk. Drizzle vinegar mixture over cucumber mixture, and toss gently to coat.
Core tomatoes; cut each tomato into 6 wedges, cutting to, but not through, bottom of tomato. Spread wedges slightly apart. Spoon 3/4 cup cucumber mixture into the center of each tomato. Serve on 4 lettuce-lined plates; garnish with basil sprigs, if desired.