Fukuoka is no Kyoto, certainly, and a couple of days are enough, but there are a number of good places to see - most are not well publicized, though you can definitely find them if you look. The Nanzoin Temple is a tremendous place, and only about 20 minutes by train from Hakata Stn.
https://www.youtube.com/watch?v=lT4VQFmHFyMAnd there are some very beautiful gardens, like Yusentei, Shofuen, Rakusuien, and at Ohori Park. Plus the sunset from Odo Park is very beautiful.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...A Fukuoka Tourist City Pass is a great way to get around:
https://yokanavi.com/en/tourist-city-pass/but many places are near a subway station, and a day pass is just 620 yen (520 yen weekends).
Seeing Kitakyushu also is a good idea, and neglected by most. Kokura Castle is not bad, the Moji Retro area preserves a Japan from a century ago, and the view from atop Sarakurayama especially around sunset is a secret jewel that even the well traveled mostly don't know.
https://www.youtube.com/watch?v=Hl1pCW9IbbkYou could even walk under the ocean to Honshu and see a bit of Shimonoseki if you wish:
https://www.youtube.com/watch?v=6y4K6erVyvAAnd if you like to go hiking, there is some great hiking in Aburayama, Shikanoshima, and Nokonoshima Island.
Dazaifu's famous Tenmangu shrine is nice, though you'll be through it in less than 20-30 minutes. The Kyushu National Museum is good, if you like museums, but 3/4 of a day is more than plenty, and I suggest you take a few hours and go see Yanagawa for the canal boats. On the way back, you could even visit some of the hot springs at Futsukaichi - which are free from the tourist crowds and have a history going back over 1300 years.
The suggestion of a 3 day JR Northern Kyushu Rail Pass is a winner, and it is easy to get you money's worth off of it. Nagasaki is one of the most interesting places on Kyushu.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-gbw2yXNLK92siHyNqSNx...If for some reason you don't want to travel so far, you could take a day trip to nearly islands like Iki for a day.
http://www.iki-island.net/Kurume has a few places as well, including the Naritasan Jibo Kannon,
https://youtu.be/G1SBtmqXteEand nearby a partially recreated settlement from the Yayoi Period (300 BC - 300 AD) at the Yoshinogari Park.
http://www.japan-guide.com/e/e8325.htmlYou should have a very rewarding time.