Sogyeok-dong 84 Jongno-gu, Seoul
http://www.wingspoon.com/seoul/samcheongdong/bukchon/index.nhn - Noodles + Dumplings! Book-chon-kal-gook-soo is the place I'd like to recommend. They have delicious Mahn-doo and kal-gook-soo. Here's the link to the Google map.
205-8 Bangi 1(il)-dong Seoul, Songpa-gu
http://www.wingspoon.com/seoul/bangidong/byeokjegalbi/index.nhn - Butcher-cut galbi; possibly very expensive and kind of touristy.
661-1 Sinsa-dong Gangnam-gu, Seoul
http://www.wingspoon.com/seoul/apgujeongdong/mandujip/index.nhn?menu=photo - Big giant thick dumplings; high ratings on Wingspoon
Songpa-gu Seokchon-dong 158-4, Seoul
http://www.visitkorea.or.kr/enu/SI/SI_EN_3_1_1_1.jsp?cid=349106 and http://www.nolboo.co.kr/english/brand/Hangari.asp - Nolboo Hangari Galbi; a chain, but looks kind of delicious
28-21 Samcheong-dong Jongno-gu, Seoul
http://www.wingspoon.com/seoul/samcheongdong/seoulthesecond/index.nhn Then, when you finish your meal, go to this place for a dessert: Seoul-seo-dool-che-roh-jahl-ha-nun-jib. Meaning: The 2nd Best Place in Seoul. Don't be fooled by the name. I believe they are the best. They're just being modest. When you ask them where the best place is, they just smile back at you. What to eat? 단팥죽 Dahn-paht-jook. A... sweet red bean porridge. Add 수정과 Soo-jung-gua after that if you're still not satisfied.