Does anyone know of a place to hire motorbikes in Thailand, say Bangkok or Chiang Mai, and ride them into Laos and back? The trip will be self-guided and should take no longer than two weeks.
Yes, it is possible to hire motorbikes in Bangkok and Chiang Mai. I haven't done this myself but some friends did. It costs about 200-800 baht per day to hire one depending on the type of motorcycle you get. It might make sense to buy one from an expat or a another traveller, then sell it when (if?) you go back to Thailand. You will definitely need to have an IDP to drive around in Bangkok as police keep a watch out for tourists.
The problem I foresee with your plan of driving to Laos could be that many motorcycle rental places in Thailand retain your passport. You could try convincing them to leave just a copy of your passport but for long-term hires they may not agree. You also need to have paperwork to cross the Thai-Lao border on bike.
Those are the basics. GT Rider is a good place to start your research on the exact documentation you need to cross the border, the kinds of motorcycles usually available for hire, maps, et al.