Major Update concerning Fukuoka (19/05/2012):
It seems that Fukuoka has now implemented several free wifi zones under the name "Fukuoka City Wifi". This started 26/04. I have no information of how stable or how fast it is. For more information, you can try this link but it is in Japanese only: http://www.city.fukuoka.lg.jp/wi-fi/.
I have just returned from a 2 week trip to Fukuoka and Seoul and wanted to share the results of my hunt for free wifi in those cities.
Seoul:I stayed only 2-3 days in Seoul and it was a last minute thing so I did not prepare anything. Knowing that South Korea is really big in terms of smart phones, Internet gaming and so on, I was really expecting to find wifi spot easily. I could not have been more wrong. All Wifi spots I found were protected by a wpa2 key and I then decided to go to the Starbucks coffee of Myong-Dong, expecting to find a free wifi spot there. On the front door, I could see something like "free Internet here", inside, I could see a lot of people with smart phones and tablets but again this was false hope. The wifi spot was depending on you having an Internet subscription with some local mobile provider and both Starbucks staff I asked told me there was no wifi. A wifi provider was even clever enough to call its wifi spot something-Starbucks so that most people thinking a free wifi spot would be available would connect to it, and may eventually have paid for the Internet (this wifi spot was not free and was not managed by Starbucks). I decided to try McDonald but before I reached it, something I had to do in real life showed up and I had to give up my hunt of free wifi in Seoul.
Fukuoka:I stayed about 10 days in Fukuoka. I had looked on the Internet beforehand for info about free wifi spots but found only 2 locations in Tenjin: tenjin chikagai and Seattle Best Coffee.
Not living near Tenjin, I tried again to find a free wifi spot by myself first and found out that Softbank (a Japanese mobile provider) had wifi spots in many places (Starbucks, fast-food, ...) but that you needed a subscription with them in order to use one of those wifi spots. They are all named FON and your mobile phone will connect to them instantly if you have the wifi on (you don't need to select them or choose to connect to them). However, as I just said, you won't be able to surf on the Internet, unless you have a subscription with Softbank.
(the sign says "Wifi available" but you have to notice the small SoftBank icon at the bottom of it, meaning it is only for Softbank customers)
Next step in my hunt was Seattle Best Coffee. Unfortunately, this shop is now closed and does not seem to reopen anytime soon (the reason may be that Starbucks opened a shop just next to it).
All in all, my last hope was the Tenjin Chikagai (underground shopping area just next to the Tenjin station). There was effectively a free wifi there (select the tenchika wifi, not the FON one) but as people mentioned in some forum, there is no bench. So you either have to stand up while surfing, sit down on the floor or find a coffee shop. You will find several ones in this area, including a Starbucks.
You will also have to deal with another problem: this wifi is a bit slow. Probably ok if you are used to it but personally I just had to give up. Pages were too slow to load and this was for normal web browsing, I just cannot imagine watching some YouTube videos there. This was the end of my free wifi spot hunt, too much hassle, too slow connections, not enough updated information about where to find them, ...
I ended up in an Internet coffee and for those who are interested, you can find it at the exit 13 of the Tenjin station, just next to a karaoke.
There are free drinks on the third floor (free drinks are common in Internet cafes in Japan), so all in all, I think it is a good option. I have also seen a lot of ads about the popeye Internet cafes. They look nice and new but I did not try them myself.