There's also a middle ground I might consider ideal: choose 4 places to go to, and spend 3 months in each. You can't pick up everything a city has to offer in 3 months (let alone a country!), but it gets you much further than the 3 days here, 4 days there strategy. Cuts down on travel and housing costs, too, both monetarily and temporally.
If you want more extended trips than that, you can often go on a day-to-weeklong vacation to a neighboring country and return to where you were before, bringing it to a 6 month stay in a city. I'd almost argue that if you want to be there longer than that, you actually should pick up a local job, not just to fund yourself (which is probably a minor point for most of us) but because it deepens your experience in a place.
If you're pretty well-off, at least some countries will offer you an "investment visa" if you can prove you have hundreds of thousands of dollars in liquid assets.