Shigaku.or.jp maintains a daily updated page of private school positions up for grabs around the nation. The job advertisements are for teachers of all subjects, but many English teaching positions are advertised in each update, and among them are listings (usually one every few weeks) for which only a native speaker is wanted.
Because the schools are private, some rules about teaching certification can be bent: A well-written resume and application coming from a foreigner will draw interest, and if you impress a school during an interview, you can bet they will find a way to hire you.
I highly recommend this site, as the positions are at prestigious, competitive schools, but they often go unadvertised among most of the foreign community. This is a chance to directly appeal to schools and get hired as a full member of staff, rather than as a dispatch or contract worker.
However, the leads on Shigaku's page are written in Japanese only, and many of the jobs themselves (even those intended for native English speakers) are often written in Japanese on the schools' websites. Conversational Japanese is a must here, and applicants with JLPT-2 or greater equivalency, coupled with teaching experience, are the ones who will have their pick of jobs.