Although I haven't gone through it myself, a number of my friends have, and it seems to me that what they are looking for is someone who will a) experience extreme culture shock, and b) bestow a degree of culture-shock upon the students. A friend of mine who is perfectly fluent in Japanese got turned away from JET for that very reason--he was fluent. It appears that they want someone who is a blank slate who has no real knowledge of Japan so that the person will learn about Japanese culture wholly through direct experience, as opposed to someone who has second-hand knowledge (study, acquaintanceships, etc).
If you get turned down, there is always the eikaiwa, but they aren't always reputable (*cough, cough, NOVA, cough*). But you can also look online for recruiting agencies that will put you in small, local eikaiwa (be careful with these, though). And you can always google "Kansai flea market" and download their newsletter, which is basically a gigantic classified ad for English-speaking (mostly) foreigners who want teaching jobs, homes, etc.