I've had a few incidents, but like John and Pete note, they're uncommon. Most of the complaints I've heard, like Pete, are from people who aren't used to riding on roads or streets, and they're scared when there's not a line of trees separating them from motorized traffic.
My suggestion is to acclimate yourself to riding on roads. Start with quiet suburban or rural streets -- most of the people who pass you will probably drive on the other side of the road to do it. After you're used to that, look for roads with a little bit more traffic. Then try a 3-5 lane road (there's one on my daily commute) that doesn't usually need more than two lanes for the traffic. (You'll want to avoid rush hour, of course, at least at first.) When you're comfortable riding there, you're ready for 97% of Adventure Cycling's routes.
Do consider looking at vehicular cycling recommendations. Start with the LAB videos, the better cycling one halfway down the page at
https://www.bikeleague.org/ridesmartvideos. Also take a look at John Allen's Street Smarts at
http://www.bikexprt.com/streetsmarts/usa/index.htm. The ideas presented at those sites freak some beginners out, but they work well.
In support of vehicular cycling, someone pointed out (and I've confirmed from personal experience) that most drivers give you as much room as you take. Ride right next to the white line? One out of ten passing cars feels like you're cleaning their window with your sleeve. Move a foot from the white line, they'll pass with a foot to spare. Ride three feet from the line, and they'll give you three feet of passing space. It's counterintuitive, but it usually works!
