Japan recognizes International Driving Permits (IDPs) issued under the Geneva Convention of 1949. (Your driving permit will list which convention it falls under.) This kind of IDP can be obtained from automobile associations in many countries around the world, here's a site to tell you what agency can issue yours. The good news is, it's really easy and cheap to get an IDP. (For example, it took me $15 and 10 minutes to get mine at my local AAA branch.)
But be careful of Internet sites offering to get you an IDP with "rush delivery" for hundreds of dollars; they're a scam, and the document you receive from them might not even be a legal IDP.
However, after you get the IDP, you need to know that individuals are not permitted to drive on an IDP in Japan indefinitely. If you have lived in Japan for more than one year, your right to use an IDP may have lapsed, even if the date printed on your IDP hasn't expired.
In principle, if more than 365 days have passed since your Landing Permission in Japan, it is illegal for you to drive on an IDP. You must obtain a Japanese driver's license.
However, if you have left Japan for a 90 day period and returned, you are again eligible to use an IDP, so long as the date on the IDP itself is also valid. But, if you left Japan for less than 90 days, then you're still subject to a 365 day period from your previous landing permit in Japan. Don't just take my word for it, check out what the Tokyo police have to say about it.
Anyway, any kind of slip up with an expired IDP combined with a grumpy policeman could result in heavy fines (up to $3000 USD), a year of imprisonment, or for a foreign resident, deportation. So, if you're going to be in Japan very long, consider dumping your IDP and getting a proper Japanese license.
As an added bonus, you'll have a form of legal ID in Japan that isn't your Alien Registration Card, so you can stop handing out your nationality and work address every time a manga cafe clerk needs to verify your identity.
Once you've got your IDP, check out our guides on Renting a Car in Japan and Buying a Car in Japan!