Robert's idea of a folding bike would be ideal. I've read about the Bike Friday folding bikes (they pack into a checkable airline suitcase), and they are supposed to be very high quality. To read about them, CLICK HERE . The advantage of bringing your own bike over renting, of course, is you know you'll be riding a properly fitted bike.

If you'd rather rent, you might ask this question on the Touring Forum at BikeForums.net ( CLICK HERE ). You might be able to pick up a bike at one city, and drop it in another. If this is possible, probably someone at that forum would know the answer.