I was going to say basically what John did. I've 'stealth camped' in places where I wasn't supposed to before with little guilt, but the east is largely posted (I reckon the west is probably a few years from this, but eastern America has an "ownership" fetish, for sure, and is heavily populated). The best option is to go 'park by park.'
Campgrounds stink in general because they are typically anathema to 'true' camping, although you can meet some really nice people. I just don't like the sardine can nature of them, the noise, the smells, the gravel, etcetera, nor do I feel comfortable sleeping in a tent while my bicycle is at the mercy of [everyone]. Add on the $15 price for such luxury, and it's just not worth it unless you really need to shower.
Asking people is always good. I have been offered so many backyard spots just from local conversations with the curious - without even asking. One option is to find where the Boy Scout lands are, and ask for a place to camp in exchange for a lecture on cyclotouring. It's so rewarding to introduce this activity to eager kids, and the Boy Scouts are friendly as a matter of course, even if they turn your offer down. Police, too, will often let you park it on their front lawn if you ask the right ones very nicely. Cemeteries are a further option.
When I was first starting this wonderful activity, I used to just camp wherever I wanted and could easily get off-road, including posted property (I wasn't cutting fences or anything, but if a place looked deserted save a "posted" sign, I'd camp there with no loss of sleep over it). In a pinch I will still do this, because I will not ride at night for any reason and will camp where I will if the sun goes down before I've made it to my planned destination - but I ultimately gave it up as a general practice for a variety of reasons, foremost of which is that it really is not fair to the landowner.
I've found that there is some measure of risk in primitive 'stealth' camping where I live (NY). Even in state parks in this area, it is not really permitted to camp au sauvage
, but again I just do anyways, and even park authorities that I've talked to after the fact were completely non-plussed and congenial about it. This, of course, is entirely empirical and not a 100% endorsement that you go about telling people what you're doing. There are enforcement-freaks everywhere, and you never know when you're going to run into one. Barring a *really* unusual coincidence or a bad choice of spots, you are never going to be found out though, and I have to admit a certain thrill in seeking your own destination.
Ken Kifer has some good words on this whole issue: http://www.kenkifer.com/bikepages/touring/camping.htm