(In Rochester) I walked into the office at Mayo and I just opened an account with them. It was very easy. I brought my passport and SSN and opened up a savings account (mandatory) and then a checking accounting (to get the ATM access), bring USD cash with you to make the initial deposit.
While I was doing it, I applied and got approved a Visa credit card and a Personal line of credit so I have overdraft protection on my savings account (it didn't cost anything but will save you lots of fees and headache if you made a mistake in the future).
The Visa credit card is given to you within the same day, for me it took about 3 hours to get my card.
Give yourself about 30-40 minutes for this initial meeting with the Mayo Credit Union.
Note that with the savings account, the first $5 is 'held' within the account as your membership fee which you get back when you close the account.
Anyone that is Mayo-affiliated is able to open an account including spouse, blood and adoptive relatives and their spouse, etc.
They are a member of NCUA, a US government agency that helps insure between $250K to $500K depending on the type of accounts so you know your money is safe.
Hope this info helps!