There is nothing un-Islamic in using credit cards as long as one does not delay paying the bills and pays the total amount on time. It is indeed haram to pay interest. However, one is allowed to use the credit cards to charge the amount that one can pay when the bill comes. But if one uses the card to borrow money on interest or to purchase what one cannot pay on time, then one is indulging in riba, which is forbidden in Islam.

Dr. Monzer Kahf, a prominent economist and counselor states: “1. The opponents of credit cards found their view on the basis that it is an interest-based loan contract; consequently, it is haram on the ground that it is riba. There is no doubt that issuing such an interest-based credit card is not permitted for any Islamic/Muslim financial institution. On that basis, Islamic banks issue credit cards that are not interest-based. Essentially, the bank withholds from your account an amount as a guarantee for the payment at the end of the grace period of the card, and each month, if your current account permits it, the credit card charges will be deducted from the current account; otherwise, the amount withheld will be reduced by the amount you owe on the credit card, and you will be asked to replenish the amount withheld. Islamic banks do not really benefit from the card itself, and they provide it as a service to their valued customers. However, they benefit from the mudarabah deposit withheld as a guarantee.
This is also the essence of the Organization of the Islamic Conference resolution on credit cards that came in the year 1999 or 2000.
2. The proponents of the permissibility of using interest-based credit cards found their view on the following:
a. While it is true that issuing interest-based credit cards by Muslims is not permissible, accepting them by Muslims is permissible because they give a choice to the consumer of either going interest-based or settling within the grace period without any interest. The contract is not a loan contract; it is rather a prior acceptance of hawalah or it is kafalah by the issuer of amount charged by the user.
b. The permissibility of signing a credit card contract and using it is conditional on two points: 1) that the user is keen and able to pay within the grace period; 2) the user will not withdraw cash (because cash withdrawal generates interest from the day of withdrawal and does not have a grace period).
In other words, permissibility is conditional on not invoking the interest clause in the credit card agreement or by using the credit card as a free rider.
3. The permissibility is not based on Muslim/non-Muslim country, nor on need for the interest-based credit cards. Many people can live without them, but they provide considerable convenience for Muslims who live in a twenty-first century society. (Please notice that all opponents of credit cards do not live in our time; they are still living in the fourteenth and fifteenth closed economy century.)
Note that the annual fee on credit cards is permissible and is practiced by many Islamic banks.”