the fastest of the Shinkansen ("Nozomi") with the JR Pass.
By train
Fukuoka's Hakata Station is the terminus of the Sanyo Shinkansen (north) and Kyushu Shinkansen (south). Sanyo Shinkansen services are offered from Kokura in Kitakyushu (20 min), Hiroshima (1 hr), Okayama (1 3/4 hrs) and Osaka (2 hr 30 min), and through via the Tokaido Shinkansen from Kyoto (2 hr 45 min by Nozomi), Nagoya (3 hr 30 min by Nozomi) and Tokyo (5 hr by Nozomi).
If you have a Japan Rail Pass, you cannot use the Nozomi, so if you are traveling from Tokyo or Nagoya you will have to take one of the two hourly Hikari trains from Tokyo and change at Shin-Osaka (alternatives are Shin-Kobe, Okayama, and sometimes Himeji) to a Sakura service. Travel time from Tokyo to Fukuoka using these trains is 6 hours.
Another option from Tokyo is to take a westbound sleeper express such as the Sunrise Izumo or Sunrise Seto, leaving Tokyo around 10PM, and then connecting to the Shinkansen at Okayama early in the morning, to arrive in Fukuoka just before 8:30 AM (or by 9:15 AM if you have a Rail Pass and use a Sakura service). While this takes much longer and costs more than the Shinkansen (from �25,000), it provides the benefit of doubling as lodging and transport.
From Kagoshima, Kyushu Shinkansen Mizuho and Sakura trains make the run to Fukuoka in 80-90 minutes at a cost of �10,170. The Mizuho is NOT valid with the Japan Rail Pass. Most Sakura trains do travel through Fukuoka, connecting Kagoshima to Osaka with no transfers.
From Nagasaki, the limited express Kamome runs hourly (sometimes twice an hour), taking 2 hr and costing �4,710 each way.
For historical reasons, Fukuoka's train station is called Hakata. If you search for schedules to "Fukuoka" online, you will likely be given an itinerary for a totally different (and much less interesting) city in northern Japan.
Overnight by train with rest stop
If you hold a Japan Rail
