Faith was a fairly well received period k-drama that had a time travelling female doctor from modern times, but the pairing of Lee Min Ho with the significantly more mature looking Kim Hee Sun felt a little strange to me. This AU fanfiction really hits the spot, taking the spunky character of the doctor and changing it to that a young healer of that era totally works without the time travelling mambo jambo and the annoying whininess of the original character.
Addendum on 27 Dec 2014:
Ok, I went back to re-read the recaps on Dramabeans. The time travelling doesn’t seem to have too big a logic gap (besides the heroine being able to always jump to the correct time in the future, but jumping back in time being so hit-and-miss), although the repetitive kidnapping and poisoning of the doctor does get annoying. There are also problems with the direction and editing, so it’s probably it’s not quite worth 24 hours of your life. Reading it to see if it’s your cuppa may well work though.
Link to the first ep here: http://www.dramabeans.com/2012/08/faith-episode-1/