Usually all banks in the US work with a debit system either with VISA or MasterCard brands. These cards are tied to the individual bank accounts and usually dont have huge fees. Especially for ATM transactions they are pretty good as Bank of America for example has big partners in about 7 countries.

This is the best way in my eyes, but forget about it to use it for car rentals or whatever requireing deposits except you hold your checking account around 2.000$ for this purpose !