Do ATM cards extracting money from US banks work in Russia? Will I have a problem using credit cards that are on US banks in Russia?
If you show up in Russia and start using your debit card at ATM's there's a possibility that your bank's detection program may flag it up for analysis. This would happen if using a Russian ATM is outside of your standard usage pattern.
At that point, the bank will start trying to call your mobile or send you email asking you to contact them, and if they cannot reach you then yes, they might put a block on your card.
To avoid something like that, you can send your bank a notification email along the lines of...
I expect to be in Stavropol during 20 July - 23 July and in Tver on 24 July - 26 July. I will be using local ATM's to replenish my pocket money and please honour withdrawals up to USD 200 per day. My mobile number is... and my email is...
Some banks (like HSBC) even have a form you can fill out online.
Other than that, there is no cause for thinking that a debit card issued by a Fed regulated bank would not work in Russia. I don't use American Express at ATM's, but can attest that Visa and Mastercard debit cards work in the cities I mentioned above along with Moscow and Peter.
To help canonicalise this answer, the same would hold true for FCA regulated banks and ECB supervised banks.