I live in Jeollabuk-do, and I agree with Omiak and SoloBuffalo. The drivers in Korea are kind of crazy, but I feel way safer cycling in Korea than I did in the U.S. because the drivers here expect me (and others) to be cycling on the roads with them, so they're much more aware of and much more accommodating to cyclists than in the U.S. I'm not Korean, so I could be wrong, but my impression is that Korean drivers don't take driving craziness as personally as U.S. drivers do - everyone drives like a maniac, and everyone expects everyone else to be just as assertive as they are, so they're all playing an equal game, rather than in America where people take it as a personal insult if they're cut off and feel the need to avenge their disrespected reputation. In America I always felt like the drivers were hoping they could run me off the road, but in Korea I feel like my right to cycle never gets called into question.
To respond to the original poster's questions (late though I may be), yes, definitely find a way to cycle here. You can buy a decent bicycle quite cheaply and there are plenty of English teachers coming and going so you can always find someone to sell or give it to when you go. It's an excellent mode of transportation and the land is beautiful to see - especially out in the provinces, you can easily enjoy urban cycling, bike paths, trips through farms and rice paddies, and really anything without going very far. It's a great opportunity and not one to pass up.