At night, vendors set up food stalls (Yatai) along the Naka River in Hakata. The most popular dish, ramen, is pleasantly made with fresh noodles. It isn't too spicy either. All of the vendors are Japanese, and a great deal of the customers are Koreans who've taken the (relatively) cheap boat ride from Busan.
Like many things in Japan, this is not a cheap place to eat. A bowl of noodles and two lage bottles of Asahi beer cost 1800 yen (~$15). But you can enjoy the river, and if you speak some Korean you can always suprise the other diners.
I was exactly there exactly a year ago -- the fourth stall. Before then I didn't think ramen could be anything special.