Is Amazon Payment the only way I will be able to receive my payment?

Bluebird (also from Amex) is free. It acts a lot like a checking account with some limitations. You can link your Bluebird account to MTurk and set up direct deposit just like with any other bank account, and payments always transfer the next day as long as the transfers take place before 7:00 PM Central Time (except weekends and holidays) in my experience.

You can also deposit checks via mobile app (though this can take up to a week according to some, but it's always worked for me in 2 business days), and you can deposit cash and checks at any Walmart store for free. You can withdraw from any MoneyPass ATM (available at most Walmarts and elsewhere depending on your location) for free and from any other ATM for a fee. You cannot deposit money via ATM, though; if you need to deposit cash you invariably have to do so at Walmart.

Ordering checks is free, but in order to write a check you have to log in to your Bluebird account and enter the check number, amount and your ATM password. The funds are automatically deducted from your account, and you receive a code to write on the check so that the recipient can verify that the transaction/funds is/are legit.

Just a heads up: If you use the Bluebird card to pay for gas, ALWAYS pay inside (NEVER at the pump) and make sure you tell the clerk to charge it as credit, not debit. Otherwise a hold might be placed on the funds in your account for up to a week or more. Also, just based on experience, pay close attention to what happens during any store transaction. If a clerk accidentally charges $20.00 for a $5.00 transaction and swipes your card, the entire $20.00 will be placed on hold until the error is resolved, which usually takes a week or more.

Customer service is not the best, but it's a decent free option for people who can't get or don't want traditional checking accounts for whatever reason.

/r/mturk Thread