I have been doing that for about 2,5 years now, as I've been working on projects around the world from Bangkok to Dubai to Washington DC to Johannesburg.
For longer stays of about a couple months in a place, sometimes I choose executive-residences, furnished apartments etc., though in a foreign country they are more difficult to arrange most of the time.
Technically, I still have a residence in New York but it is always on sublet and apart from a few trinkets and books, I've nothing left there. I use my company's address for everything and it gets forwarded halfway around the world just fine.
One bottleneck is that your passport's visa pages tend to run up pretty fast - especially you are always on a foreign land wherever you happen to be and every entry and exit is meticulously stamped, annotated etc., not mentioning the countless visas that take up the whole page: I am almost on my third passport in the last three years, and this despite the Turkish passport being somewhat larger and having more pages than a comparable U.S. or Canadian passport. btw, somewhere around this site there must be a thread searching for the most passports used up in shortest time
)
I actually have found it most difficult to answer the question "where are you from?" in a bar. Do I say Bangkok, 'cause I've been living there for over a year now? Do I say New York, because that is technically where my headquarters are? Or do I say Istanbul, where I was born and raised?
I guess I'm a true gypsy...