You are required to report your winnings in the same way you are required to report your tips: the law is explicit, but it's very difficult to enforce and most people underreport.
Incidentally, I asked this question a month or two ago, and I figure some of the people who helped me in that thread will appreciate this update: there is nothing wrong with TurboTax. I've been using TurboTax for years, and I had stated that TurboTax had omitted winnings under $1,200, but I was wrong. I looked over my 2005-2007 returns and my winnings were included as income; I just never realized it because there's no separate sheet or breakdown of wages, earnings, and tips - when I looked over my W-2's, I noticed a discrepancy between the sum and my gross income, and realized the difference was exactly my winnings. So no 1040X's for me!