For readers hoping to find a secret, easier way to work in Japan than finishing college, I'm sorry to say that these other options aren't going to thrill you with their ease:
Immigration in Japan generally expects work visa applicants to have a 4-year university degree. However, there is a 5-year, full-time work experience qualification that applicants can meet in lieu of a degree. The work experience needs to be in the same field as the career you intend to pursue in Japan, and you'll need to be able to document the experience in your application to Immigration. Sponsoring companies hiring English-speaking individuals hailing from the US, UK, Australia, etc. will often assume you are applying for their jobs with a 4-year degree in tow. If you are not and hoping to apply for your work visa based on experience, be up front about it with your company (they may not approve) and expect your application to receive a more critical eye from Immigrations.
Working Holiday Visa
Some countries have a special agreement with Japan that allows youth aged 18 to 30 to come to the country for up to one year (or rarely, a year and a half) with the primary intent of an extended vacation in Japan. During this vacation, you are allowed to engage in work to support your lifestyle. There are restrictions on the kinds of work allowed, but with an otherwise strong application, this visa is available even without a university diploma. It is possible to start out in Japan on a working holiday visa then turn that into a normal working visa, spouse visa, etc. However, the normal requirements of the new visa (university degree, Japanese spouse, etc.) still apply.
Exchange Student Visa + Part-Time Work
If you are already in Japan as an exchange student, it is possible to extend the permissions granted by your visa to include up to 18 hours per week of part-time work. To do this, you will need signed documentation issued by your university that approves your intent to work. Requests like these are usually only approved for students working at accredited 4-year universities. Students at Japanese language schools (that is, schools whose primary purpose is to teach the Japanese language to non-Japanese) are not likely to get the extension.
If you are married to a Japanese spouse, you can apply for a spouse visa, which will allow you to work unrestricted in any industry and company that will hire you. However, without any history of a work visa in Japan, your spousal visa application will be scrutinized by Immigration. You may not be approved for a long Period of Stay, meaning you will need to renew the visa frequently. And, your spouse will control your ability to be in the country; if you at any time become unmarried, or if Immigration loses confidence in the stability of your relationship--even if you didn't actually sign for divorce--you will be unable to renew your visa. If you get a divorce with time remaining on your spouse visa, Immigration expects that you will report the divorce. However, you can continue to reside in Japan and apply for a new visa status with the time remaining on your spouse visa even after the divorce.
The Millionaire Option
If you have capital and connections enough to start a real company, with employees other than yourself (to whom you are paying a salary), there exists a business manager visa. This route requires cash-in-hand of a sum not dissimilar to the full tuition of a four year public university degree (one Gaijinpot article suggests ¥550,000 plus a generous buffer). And on top of that, you will need help from Japanese lawyers to establish the company, purchase real estate, write employee contracts, enroll in appropriate insurance programs, and apply for the visa on your behalf. As long as the business exists, whether or not it ever reports a profit is irrelevant in terms of being able to obtain and renew the visa. However, at the end of the day, the visa is only valid for working at the company you just set up.
I hear some of you asking, "But, what about permanent residence?" Well, yes. If you have permanent resident status in Japan (or Japanese citizenship), you can work in any company that will hire you. But, the gateway to permanent residence is almost always through years of renewed working visas or spousal visas. In principal, if you have been living in Japan for five years or more and are married to a Japanese National, or if you have been living in Japan for ten years or more and held a valid working visa for at least five of those years, you are eligible to apply for permanent residence. Submission of supporting documents and interviews with Immigration follow the application to determine whether your request is granted or not.
Prisoner of War Exception
The other, quick route to permanent residence is to be imprisoned in North Korea for 39 years and wanted in your country for military desertion.
Applying for Japanese Citizenship usually, but not always, comes after maintaining permanent residence for a few years. Japan does not (yet) support dual citizenship, and if you obtain Japanese citizenship, you are asked to revoke your other citizenship(s) at respective embassies. (But, it's not like the Japanese authorities tear up your old country's passport in front of your face; I have heard many cases of people secretly maintaining their old citizenship just by "forgetting" to report to the embassy and revoke it.)
So, that about covers it. Yes, if you want to work in Japan without getting a university diploma, there are, technically, other options for doing so. If I've missed any other big ones, or you have any other questions, please leave a note in the comments.