It's incredibly important to have someone you can trust to take care of admin issues at home if you're gone a while though. My parents helped me out, but a lawyer or accountant will also work - but cost you a fortune. Possibly you can hire a temporary admin assistant, but make sure you trust that person to look after everything while you're gone; you never know what will happen.
The only problem that I've found has been getting a reliable internet connection for a couple of months at a time - if you have to work on sensitive things or you want to work on things for a long time, you don't want to work from public hotspots. I'd love to get some advice for that situation myself.
Or can rent a apartment that comes with it's own landline internet connection. The landlords can set it up in their name for you and it would be part of your rent.
Santiago: ~USD$250/mo all inclusive (with flatmates) Buenos Aires: ~USD$480/mo all inclusive Studio
The idea about the net cafe is a nice one though - maybe I'll try that next time! :-)
The food and transportation is very affordable, and really really cheap if you don't go to the touristy restaurants and instead head to el mercado (marketplace). You can get a full meal with soup, drink and dessert for like US$.60-$1
Now, if you want to meet fellow hackers/entrepreneurs and recruit incredible talent, check out http://limavalley.com the startup/tech community in Peru.
What I've done is just travel as tourist staying in each country around 3 months.
When I "lived" in Chile I just took weekend 'visa vacations' to Argentina and had my stay extended for another 90 days.
Nothing changed in terms of taxes for me, since I never really become resident of another country.
Earning money online definitely makes things easier since you never have to be paid as a local in whatever country you are at, therefore, not working there illegally or anything like that.
I am not sure how much money your online business makes you, but I usually aim for developing countries, where you can easily live on US $800 - $1200 tops a month. Like bradleyjoyce said I would recommend Latin America. I've lived/traveled/worked in Colombia, Chile, Peru, Argentina, Mexico.
Also, very important, make sure you have someone reliable back home that can help you deal with administrative/misc., stuff like depositing a check, receiving important mail, faxing stuff, and the sort of things you may not be able to do online.