Only by credit/debit card. The regulations exclude checks, so most sites have an 'echeck' option.

I believe there were quite recent new regulations which stop all bank payments from US banks...my bank sent me a letter (although I'd never paid one of the sites) about this, at which point I asked the player.

Am I mistaken about what happened?

If your bank is aware that a payment is going to an illegal gambling site, they are required to block it. However, the regulations don't require them to scrutinize checks since it's just not practical.

Merchants who accept credit/debit cards must code the transaction for whatever their industry is, so those in the gambling industry are easy to block and banks are required to do so.