When I was at Laramie from 73 through 96, we used chains, but we kept them tight. The pilots who had problems were the ones who left them loose, so that the airplane would rock in the wind and suddenly pull the chain taut. That sudden jerking stop will damage things, for sure.
I saw a beautiful 170B virtually destroyed that way. The guy came in, the weather was pretty calm at that moment, and when he tied down (wouldn't allow the ramp rabbit to touch it), he left the chains sagging. When he walked into the FBO, I suggested he might want to snug up the chains--he replied with one of those "expletive-deleted" comments that made me think he would deserve whatever happened.
Overnight, the winds came up, and when I came out to the airport the next morning, that poor 170B had a bent wing, bent strut, obviously broken spar. I later heard that the guy had no hull insurance, though I don't know that for sure.
I use nylon straps now, for convenience, but I still carry nylon ropes, which I'll use in any real blow. When I use them, I use what some people call a hurricane knot, and what I've always called a running lock knot. I learned this from an old boy that ran the Evanston, WY FBO some 40 odd years ago, I've used it ever since, and it's rock solid--no slipping at all. It's basically a double overhand knot, which on the second wrap is pulled tight and locks itself, followed by a second double overhand knot which locks, with the tail of the rope then tied in a third overhand knot and snugged up to the lower double overhand knot.
Thinking that damage will occur by snugging the tiedowns tightly is mostly an old wives' tale. Normal category airplanes must be able to handle negative G loads of 1.52; utility category airplanes must be able to handle negative G loads of 1.76, both with a safety factor of 1.5. Unless you use a come-along, the likelihood of being able to put a load onto the wings equal to 2.28 negative Gs (normal) or 2.6 negative Gs (utility) is pretty slim. Certainly you can't do it, just by pulling down on a single rope threaded through the tie-down ring--no matter how strong you are, you can't put more tensile force on that rope than your weight, and for most of us, it's a whole lot less than our weight. So if you assume that the pull on the tie downs of a 1500 lb. airplane is equal between the two wing tie down rings, factoring in the negative G loads they're designed to take, that means that you could theoretically put a 1700 lb. maximum pull on each one of them without damage; a constant pull of let's say 100 lbs would be less than 10% of the normal category negative G force load allowed without regard to the 1.5 safety factor. Hard to think that could ever cause any damage, as long as the pull is constant. Even much heavier pulls caused by rocking would be unlikely to cause damage, if there's some give in the tie down ropes.
Using nylon rather than hemp or some other natural fiber rope has the added benefit of some stretchiness, so that in a gusty wind, the nylon fibers absorb some of the rocking energy. New nylon 3 strand rope will have a working strength of around 475 lbs--but a breaking strength of over 5600 lbs (that' not a mistype--5600). Even 3/8" nylon rope will have a working load of 270 lbs. and a breaking strength of over 3200 lbs. However, if nylon is left outside constantly, it will gradually lose strength, so it should be replaced periodically. Chafing can be a problem over time, so just like using thimbles and chafe guards with anchor rodes helps reduce that, similar methods can be used to reduce the chafing of tie downs.
I also agree, though, that in a heavy blow, there's nothing like being there to do whatever is necessary to keep things snugged. I worry more about someone else's airplane blowing into mine, because they didn't use good enough tie downs or adequate tie down ropes, than that mine will come loose.
Cary