The best way to do it would be to get a Sanyo Sanin Area Pass - not the JR Pass, then a 5 day JR Northern Kyushu Pass. There is just one snag - first, the Sanyo Pass is only good for the Sanyo line in Kyushu, which ends at Hakata Stn. From there and then (April 15), or from Kokura Stn, in Kitakyushu, you can start using the JR Northern Kyushu Pass. But that leaves you with one day of travel uncovered. Fortunately Kurume is not worth a full day, so on April 19th, you could spend a day or half day in Kurume and then in the evening make it over to Fukuoka for the last day of your Kyushu Pass.
Another option since you want to go to Kurokawa which requires a bus is a Sun-Q Bus Pass:
http://www.sunqpass.jp/english/route/index.htmland perhaps a 3 day JR Northern Kyushu Pass.
There is a ton of good places to see.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide:
http://www.thejapanfaq.com/fukpic.htmlThere are also many useful videos to give you a preview:
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...Look into a Fukuoka Tourist City Pass also:
http://yokanavi.com/access_tourist_egBe sure to try the tonkotsu ramen at least once, and see the Nanzoin Great Buddha:
https://www.youtube.com/watch?v=lT4VQFmHFyMIn terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind.
http://www.ngs-kenkanren.com/eng/http://www.at-nagasaki.jp/foreign/english/http://www.youtube.com/watch?v=QdWRyHGW3rkBe sure to see Mt. Inasa at night if the weather is clear, and try the Nagasaki chanpon and saraudon! Also the Koshibyo temple is an often overlooked shrine that is quite unique in Japan:
https://www.youtube.com/watch?v=Mu_JnkVInTUBeppu is famous for its hot springs, Hells Tour, plus monkey park & aquarium. See:
https://www.youtube.com/watch?v=K2NNi8MLcXMhttps://www.youtube.com/watch?v=zepPfkuqhokhttps://www.youtube.com/watch?v=cBnZneDsKqQKumamoto is famous for its castle, Suizenji Garden, a must-see gem.
http://www.visitkumamoto.com/http://www.pref.kumamoto.jp/english/pdf/kankou_e.pdfhttps://www.youtube.com/watch?v=K9xyXmU8U1gFor Kurume, be sure to see Naritasan, a hidden jewel in plain sight that few venture to see:
https://www.youtube.com/watch?v=G1SBtmqXteELastly if you have time, Kitakyushu has a few sights of interest also. It's Kokura Castle is one of the better ones, the view from its Moji wharf and preserved buildings are quite historical, and the view from atop Sarakurayama is one of Japan's best and rarely mentioned:
https://www.youtube.com/watch?v=Hl1pCW9IbbkFor comprehensive info on Kyushu, see:
http://www.visitkyushu.orgEnjoy your trip!
