Working remotely for a foreign corporation generally isn't against the terms of a tourist visa (consult the specific country you're visiting). All officials I asked had no problem with it, from South America/Asia/Europe (Germany will actually let you stay longer, up to 2 years, if you're a remote worker/freelancer). You do need a work visa if you're planning to do work for a company that exists locally (competing with the locals, etc).

It was hugely income tax advantageous. There's a huge US tax credit if you aren't living in the US (your first $95-120k tax free), and I wasn't staying long enough in any country to meet their reporting requirements.

There are several countries that will offer you citizenship by investment, including some with visa-free entry to US and Canada. Of course there is no guarantee this will last or that you personally will be able to travel (see Roger Ver).

Visas would take some planning for countries requiring a visa for your passport. Many cities have consulates that you can go to. For example, if you were in Tokyo and wanted to go to Bankok next and needed a visa for Thailand, you'd head over to the Thai consulate near Meguro station to complete the formalities.

I suppose I may be skirting local employment laws in some places, on the other hand, my employer is not likely based in that country and would therefore have no tax reporting obligations.