Other than the bibimbap, the region is famous for its hanjeongshik, which comes with two soups, 13 banchan, four or five vegetable banchan, and a salted fish.
In Muju, try the Jeonju Hangukgwan, 418-19 Deogyu-li, Seolcheon-myeon, Muju-gun (tel. 063/322-3162), open daily from 7am to 11pm, or Seommaeul, 1357-1 Eupnae-li, Muju-eup, Muju-gun (tel. 063/322-2799), open daily from 9am to 10pm.
In Iksan, I found the Geumgang Shikdang, 568 Wungpo-li, Wungpo-myeon, Iksan-si (tel. 063/862-7000), to be quite good. They're open daily from 9am to 9pm.
In Gunsan, if you like blue crabs, try Goongjeon Ggotgyejang, 23 beonji, Geumseon-li, Oksan-myeon, Gunsan-si (tel. 063/466-6677 or 8), with meals ranging from W20,000 for a raw crab meal to W45,000 for a steamed crab meal. They're open daily from 10am to 10pm. If you're looking for a cheaper meal in Gunsan, Jayu Sigan, 742-1 Jochon-dong, Gunsan-si (tel. 063/452-7522), has kal gooksu (hand-cut noodles) starting at W3,000, daily 11am to 2am (they close the second Sun of every month).
Outside of the national and provincial parks, there are small restaurants at the entrances, where local owners serve dishes from the region. Ask the ajumma (married women) what the specialty of the region is, or you can't go wrong ordering the hanshik, which is short for hanjeongshik.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.