Banking in Mexico can be tough, especially for us gringos who are struggling with the language.  Most banks have a terrible reputation as having very poor service, but fear not, there is help on the way.  Some history, until NAFTA became law in 1994, it was illegal for a foreign corporation or person to own a bank in Mexico.  One of the provisions of NAFTA required that Mexico open up their banking industry to foreigners.  Although Mexico put up a good fight to oppose this, eventually foreigners, most notably from Spain, United States and Hong Kong took over the industry largely closing out the Mexican who could not compete.  Unfortunately for many of us, this deregulation has lead to increased fees and not a whole lot better service.  Word to the wise, shop around.  Find a bank that caters to Americans.  BBVA Bancomer (Spanish owned) has what they call a preferred customer unit, a division that specializes in servicing the banking and financial needs of foreigners in Mexico.  Their entire staff speaks English. 

It's a common misconception that you need the immigration status of FM3 in order to open up a bank account.  This is one step up from an FMt (tourist) and can signify anything from you are permitted to work to you are not permitted to work but you may stay there longer than the tourist visa of six months.  When I opened my account in Mexico I needed an FM3, but it appears things have changed.  Most foreigner friendly banks, such as Bancomer, are only asking for proof of at least part time residency, such as a untility bill to open a peso account. 
Shop around, find a bank that is friendly.  Happy Banking....