I happen to help run another forum, non-poker related, that involves affiliate marketing and earning cash and prizes for trying product trials, surveys, etc. (like the free iPod giveways you always see in ad banners) and this question comes up there a lot.
I'm new here and don't know CC's policy on such things, but at our forum we always start first with the disclaimer that you should never seek tax advice from an online forum, least of all one that isn't tax-related. Ask a tax professional before deciding what is appropriate for your situation.
Now, with that disclaimer out of the way, I always take the high road and say that you claim everything you win. That much is simple. Legally, you are required to, so there is really no question there. In the affiliate marketing industry, many of the companies that ship cash/prizes get a W9 from you and send you (and the IRS) a 1099-MISC at tax time, particularly if it's over the $600 1099 threshold. However, whether a W9 is gathered or a 1099 submitted, or whether it's above $600 or not, the fact of the matter is you are legally required to report ALL income. Now I realize that given the certain US situation with online gambling, and these sites being located overseas, you aren't going to get reported by them -- however that doesn't absolve you of your legal responsibility. All it takes is for the IRS to get a red flag (a report from someone, your bank, etc.) and audit you. It really isn't worth the risk of committing tax fraud to try to hide your winnings.
Now *how* you claim it might be open to interpretation. Some claim it as hobby expenses, if it's not your primary source of income. If it is your primary source of income, then you're considered self-employed and will have to file Schedule C (and pay the full load of SS taxes in addition to the income tax). However you probably gain the ability to write off certain expenses related to your "gambling profession." Here's where it can dicey though, and where you really should consult a professional if you have any doubts or questions.
Personally I always prepare my own taxes with software, like TaxCut, and it specifically asks me during the process if I have any gambling winnings, so if you use such software that's where I'd probably recommend you do it.