Damn near impossible to do in 2012.
To be able to cross borders, you don't just have to buy a bike. You also have to have registration in your name, insurance in your name, title in your name, and sometimes a carnet depending on the country.
All of those things are a huge PIA to do on a tourist visa in most countries.
I've purchased bikes in other countries and it works fairly OK sometimes - depending on the country - and it can save you money for in-country riding. But trying to cross borders is a logistical nightmare.
I think the only place where this might work out OK is the Eurozone... I know some foreigners manage to do it and tour Europe this way.