Korea is one of the most wired and wireless friendly countries in the world.  With speeds for internet in excess of 40 mbits down and 8 mbits up Korea is an internet junkie's dream.  Hence, my first recommendation would be Skype.  It is free Skype to Skype calls, and Skype to most numbers in the world is about 2.1 cents a minute.  There are a number of portable phones (mobile) that can be rented, and incoming calls are free with most calls dialed being charged at 300 won (30 cents) per minute.  Not all public phones can be used to call internationally, and calls from public phones to mobile phones will cost more than just to another local number.  Hotels will charge you a "service" fee often for using a calling card although this can be negotiated at times.  Again, I think your best bet is Skype or renting a cell phone for your time in Korea.  Again the incoming calls are free (just get everyone else to call you)!

As of July 2012, cell phone rentals are less than $3/day plus outgoing calls. For some reason, rental stores are called roaming centers. They are everywhere in Korea. However, you need to return the phone to the same store that you rented the phone from. Incheon Airport has a number of phone rental kiosks. It was very easy. You could even rent a Korean cell phone before you board the plane in Los Angeles. They advertised that it was cheaper than renting in Korea but this not true. Also, to save money on SMS use a free SMS service to South Korea such as SMS4all.