The first thing to do is reaching Hakata Station, in Fukuoka prefecture: you can get here by Shinkansen from Tōkyō and Ōsaka.
Dazaifu and Tōkyō are more than 1.000 km far from each other, the journey takes more than 5 hours and costs 22.000 yen.
From Ōsaka it is possible to reach Fukuoka prefecture in 2 hours and 40 minutes, at the price of 14.000 yen.
Thanks to the low cost companies spreading also in Japan, you can find tickets for 7.000 yen from Tōkyō. From Kansai International Airport, by booking in advance, you can fly to Fukuoka at the price of 2.000 yen! You can then reach Dazaifu from Fukuoka airport in an hour.
At the airport railway station, take the metro to Tenjin Station. Once you’re there, take the JR west Ōmuta line to Nishitetsu-Futsukaichi Station. In five minutes you can then get to Dazaifu Station, which is the last stop of the Dazaifu line.