Peel turnips; cut into bite-size pieces. Cut broccoli into flowerets. Peel and slice carrots. Boil all in lightly salted water for 3 minutes. Drain. Place in serving bowl with red pepper and red onion.
Heat olive oil, cider vinegar, light-brown sugar, salt, cayenne and ground cumin briefly in saucepan. Pour on top; stir. Cover; refrigerate overnight.