Don't know what it's like in Korea, but here's some thoughts from living in Taiwan:
- There's no such thing as free currency exchange. Try buying something in Korea with your US credit card and returning it immediately after. You will not get the same amount back. If you only go in one direction (e.g. pay non-US expenses on a US card, and then pay off that card with USD) then the cost might be small, and it might not be obvious, but it's still a cost -- although it's the best you can do in that situation. But adding more transfers adds more layers of costs.
- Check out what other benefits are offered by your local credit cards. For example, I have a card that gives 40% off movie tickets at the local theater chain. Obviously I'm not going to turn down a 40% discount just to get some extra Citi points. Another card gives free parking at department stores, which is also worth a lot in the city. 10% or more off popular local restaurants. Discounted hotel rates when traveling domestically. Etc.
- Check out what benefits are available on Korea (or Asia) based airlines. Co-branded credit cards can give ticket discounts and status benefits (separate check-in lines, lounge access, priority baggage/boarding, etc.) All very helpful if you travel even a couple times a year.
- After all that, how much is left for spending on your US card? In my case, not very much. Sometimes I'll put foreign travel on a US card (Capital One Quicksilver -- no annual fee, straight cash back, don't have to think about it). But recently there have been some new cards in Taiwan with attractive cash-back/rewards even for foreign purchases, so I'm starting to do more of that these days and avoid the trouble/cost of replenishing my USD accounts.