There are other ways. One which used to work but which I haven't tried recently is to open a brokerage account -- which have very lax verification requirements -- with ETrade or one of the other international low fee brokers. They're quite used to having folks abroad open accounts to invest in the US markets. After you have a brokerage account, ETrade knows you for the purpose of KYC, by reference to your "pre-existing business relationship". Then you call up ETrade and say "Hey, you're also a bank. I'd like a checking account tied to my brokerage account." Bam, done.
After you have one American bank account getting a second one is a cinch, incidentally. (cough ING Direct cough.)