Actually, I'm not sure that they have been eating whale for centuries. I think whale-meat took off to some extent during WWII, due to food shortages.
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HISTORY OF THE TRADITIONAL DIET: JAPANESE AND THE WHALE
(from "Whale and Traditions of Diet", 1987)
According to a noted culinary researcher, the late Mr. Tetsunosuke Tada, Japanese probably started to eat whale during the Jomon period. This section will focus on the Japanese traditional diet with relation to the whale since that time.
1) The Jomon Period (7000/8000 to 3000 B.C.): whale as an essential food
There are a few facts that can be obtained as historical evidence which provide an explanation of the Japanese consumption of whale in this period. The first facts can be found in one of the traditional songs of the Kushiro Ainu and in the lyrics of an ancient songs called 'Yuukara' from the Saru Ainu (a group of Japanese aboriginals who have lived in the northern part of Japan since the Jomon period).
The song of Kushiro Ainu described the following story: "There was a standard whale on the Toya coast. A young fellow found it. He shouted the news around from village to village." Although Toya is currently located about 12 km from the shore line, it was on the shore about 4,000 to 5,000 years ago when the Kushiro Plains was still a part of the ocean. Several pieces of Jomon (straw-rope pattern) pottery were found in Toya shell mounds.
In the 'Yuukara', whales are mentioned: "Killer whale, god of the ocean, please bring more than one and a half whales every year. Then, I will be pleased to give my sweet daughter as your bride." As can be seen from this ancient song, the whale was a very important source of food for ancient people of Japan.
The other historical evidence can be found among the objects excavated from shell mounds. Shell mounds have been found all over Japan and provide much information about ancient people's diets. They contain bones of deer, wild boars, whales, dolphins, sea lions, fur seals and so on. This indicates that the people of the Jomon period ate whales.
Then, how did they catch large whales? They were able to hunt most mammals and animals except whales by using bow, spear and gaff. Could they hunt whales with these tools? One picture which was found from a shell mound solved this question.
http://luna.pos.to/whale/jwa_trad.html