Cook pasta according to package directions; drain. Rinse with cold water; drain again. Transfer to a large bowl. Stir in tomatoes, cucumber, green onions, and olives.
For dressing, in a screw-top jar combine oil, lemon juice, basil, snipped or dried oregano, garlic, salt, and pepper. Cover and shake well. Drizzle dressing over pasta mixture; toss gently to coat. Cover and chill for 4 to 24 hours.
Before serving, sprinkle salad with cheese. Garnish with oregano leaves.