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.
Add the radishes to the salad just before serving. Otherwise, their red color will bleed onto the pasta and vegetables.