Not directly relevant to the question, because you're asking about the USA. But in the UK, I believe you can set up an airfield anywhere you want as long as the landowner gives you permission. The only problem is, you can't use it whenever you want.
There's a rule in the UK called the "28 day rule", which means that you can use your private land for any (legal) purpose at all, but only for 28 days a year. The rule was created to enable people to run car-boot sales, etc, from their private property without needing explicit permission - but by restricting it to 28 days a year, they ensure that you can't set up a permanent business. (What's the American equivalent of a car boot sale? I assume Americans don't use that term, because you guys call a boot a "trunk".)
The same rules will also allow you to use your private property as an airfield, but only for 28 days a year, without needing to get permission from anyone.
Of course, there are low flying rules which you have to be able to comply with. Some of the low-flying rules contain exemptions for take-off and landing which apply only to licensed airfields, so you'll still have to obey these rules when taking off or landing at your private strip. This will limit your options somewhat. And safety concerns will obviously play a big part in deciding where to set up your strip.
This is my understanding from bits that I've read here and there - I'm not an expert, and I've never researched it since I don't expect I'll ever own enough land to be able to do this myself! But I thought it might be interesting.
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