Prepare pasta according to package directions, omitting salt and fat; drain. Rinse with cold water; drain.
Combine pasta, broccoli, celery, and water chestnuts in a bowl; set aside.
Prepare dressing mix according to package directions; stir in oregano. Pour over pasta mixture, stirring to coat. Cover and chill at least 6 hours, stirring occasionally. Just before serving, stir in radishes, and sprinkle with cheese.
Tip: Add the radishes to the salad just before serving. Otherwise, their red color will bleed onto the pasta and vegetables.