This might be insane, but I'm thinking about attracting online retail deposits (CDs only) from residents of countries with very low or negative interest rates (Denmark, Greece, etc.) What do we need to worry about besides AML and BSA?

i think your first question should be to bank counsel that has strong knowledge of items of this nature...such as regulatory, advertising, CIP, BSA, and many more...

Tax reporting and related interest withholding issues would have to be studied and nailed down. The U.S. has tax-reporting treaties with some countries. This will add complexity to your year-end processing.