Dazaifu shrine is well known for plum blossoms. They bloom from mid-February to early March. There are 6,000 plum trees on the grounds of the shrine. When they are in full bloom, they are gorgeous.
Plum trees were introduced from China (Tang Dynasty) to Japan in the 8th century, and they became the favorite of many Japanese artists - there are many ancient poems about plum blossoms written in the 8-9 centuries - many of them were written in Dazaifu by famous ancient poets.
I visited Dazaifu in late February of 2011; they were only about 10-20% in bloom. In other years they are in full bloom by late February. It varies from year to year. I don't know of good source in English for the information. Dazaifu shrine has an official website in English and other non-Japanese languages (http://www.dazaifutenmangu.or.jp/en), but it does not offer info of when the plum trees will be in full bloom - I wish they improve their website. You may want to ask your Japanese friends or acquaintances.
There are several rest houses on the grounds and many places to eat on the approach to the shrine. There are not many signs in English, but they have pretty good pages in English, PDF brochures and maps in English in their webiste (URL above)
Dazaifu gets extremely crowded on New Year (Jan 1 - Jan 3), but in February it's not too bad.
There is a very good hot spring, Futsukaichi Onsen, very close to Dazaifu. You may want to combine your visit to Dazaifu with a stay in Futsukaichi hot spring.
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