Since the mid- to late-1800's, Japanese record-keeping has been done through the Koseki, or family registry system. I was actually born in Japan, so my koseki is currently filed in Ryogoku, Tokyo, where my parents lived at the time. This form also serves as a birth certificate, and is what citizens need to get a passport, etc. The first and most basic step in doing family history research in Japan (aside from just gathering what information you may already have from family members) is to request copies of koseki from the municipalities where your family comes from. There are a few sites that provide forms that you can fill out in order to request copies. You will probably need to attach copies of documentation showing that you are a direct descendant of the person/family whose record you are requesting. In my case, that meant sending a copy of my passport and a copy of my own koseki. If you aren't a citizen or your ancestry is more far-removed from Japan, it may mean additional documentation to show who your parents/grandparents are. Not to mention that you'll have to include a money order for the fee. I did this while I was in Japan, which made it easier, but as soon as I figure out how to do it from the states, I'll let you know =).
On a recent trip to the LDS Family History Library in Salt Lake City, a couple of sweet Japanese ladies told me that they recently changed the rules regarding koseki so that you could request the record of someone who is not your direct ancestor if you have permission from that person or from a descendant of that person. This may open up the range of records that you can get, and I am excited to see if I can start doing some research on my step-grandfather's family or on various aunts and uncles' families. But I only know what the ladies told me, and they weren't sure how the changes would eventually play out.
Another catch in the system: all the forms are, of course, written in Japanese. An they are generally written by hand. And they often use fancy or antiquated characters rather than the standard characters that you may have learned in an elementary Japanese course. So you will probably need to find someone fluent who is willing to help with some translation. I'll try to eventually post some tutorials on figuring out names and dates, etc.
Pretty soon here I'll update this post with some links to further explanations about Koseki and instructions on how to request them.
Koseki on Wikipedia
Rootsweb Link List
Translating Koseki (for translators, not amateurs, but may be helpful?)
Koseki on Nikkei Ancestry