How long are you going for? my guess is that it would be much cheaper to buy a bike than to rent one. if you don't mind riding a piece of ****, you can get a mountain bike anywhere that will get you around for between about $150 -$300. a semi-good bike more suitable for touring(though not a true touring frame) can be gotten for around $500-$700. I've recently been looking around for a touring bike for my daughter. she's 12 so i don't want to spend big bucks on a bike she'll probably still outgrow. a cheap touring bike goes for around $700 - $800 or so. the only issue for you if you want a decent touring bike would be sizing issues, especially if you're anything taller than about 5'10" or so. you'd have to do a lot of looking around or wait for a special order.
here's a less expensive aluminum frame tourer with good wheelbase and chainstay clearance. the brand's name is riteway. look for the shepherd 1.0.
http://www.riteway-jp.com/riteway/index.htm
here's another: steel frame and a number of nice details like lugged frame, braze on for pump(which comes with bike) etc. main downside is the 26 in wheels. look for the emperor touring master
http://www.maruishi-cycle.com/
fuji makes a decent not-too-expensive tourer, but i've yet to be able to find their japanese homepage. here's their u.s.
http://www.fujibikes.com/2005/bikes.asp?id=22
however, why bother to buy one after you get to japan? you'll be wasting time and not saving any money. a boxed up bike is only 1 piece of luggage. you and your wife can throw your panniers in another box and still be way under the baggage limit. speaking of panniers, buy them in the u.s.. there's a very poor selection and they're outrageously expensive.
since you won't be needing a tent, stove, etc., 2 decent sized rear panniers and a handlebar bag sound like they would be sufficient.
if you're going rural, almost any little village would have a "minshuku" if you get stuck anywhere. however, it is well worth it to get a youth hostel membership. get the handbook and map for japan. there's got to be about a thousand or so of them, and they're great for meeting people.
good luck and have fun