I can only speak about Estonia, but payment by card is extremely popular -- more popular than in the US in my experience. Most people have a Mastercard or Visa debit card. Banks often charge a fee to have such a card, of around 1 euro/month.

Credit cards are not as common because banks have stringent requirements. I once asked for a credit card for my Estonian company from the bank we've used for years, with a 500 euro limit, and that had to go to a credit review committee and was nearly rejected. All this despite our long banking relationship and a history of having a balance of 50k euros or more at all times with that bank.