This is cool but not a part of any standard curriculum in Japan despite the title.
What I found greatest about the Japanese method of learning multiplication was actually the method for learning single-digit products. The Japanese use a system called "kuku" (translated, "9 by 9") which involves memorizing a rhythmic chant that goes through the entire multiplication table, with each product being concisely expressed in a few syllables. This is made possible by the various ways in which a number can be pronounced in Japanese.
I think the method is made necessary because to say the full products takes a lot of syllables in Japanese (e.g. 7x7=49 would be "nana kakeru nana wa yonjyuu-kyuu"). So perhaps it doesn't differ too much from the way you learn multiplication tables at an English-speaking school, but a cool method nonetheless.
Video here: http://www.youtube.com/watch?v=1hLyzXM53IE
I've always thought it very valuable that they do math both there and at their regular US school in the hope that it will mean they aren't dramatically slowed down when math appears in a conversation like second language learners are. They certainly are crazy fluent in both languages otherwise compared to any second language learner, but I recently discovered my oldest tends to think of multiplication products in Japanese first even when in an English context. The brain is tricky :-)
seven [2] times [1] seven [2] equals [1] forty-nine [3] = 9
七[1] 乘[1] 七[1] 如[1] 四十九 [3] = 7 (without using primary school mnemonics)
七[1] 七[1] 四十九 [3] = 5 (using primary school mnemonics)
Even at that primitive level, a change of language can give you a 23% and 45% speed-up of basic arithmetic operations, respectively.
English, 1 digits: 6.5 syllables; English, 2 digits: 14.9 syllables; English, 3 digits: 25.4 syllables.
I'll let other people figure out the numbers for their own languages, but I'm guessing most languages will have similar lengths. Numbers tend to be short in any language.
All I can say is that in my personal experience learning the multiplication tables in both languages around the same time, it was dramatically easier in Japanese. Perhaps the small reduction in syllables caused that, and perhaps the "rhythmification" was more responsible.
I do recall seeing a study somewhere that showed people who spoke Chinese were able to remember more numbers, ostensibly because each digit takes less syllables to speak in Chinese so the total syllables to remember in one's head is smaller.