I one time put my Quickbeam unboxed/unbagged on a flight to Carlsbad, CA by way of PHX from Milwaukee. The plane between PHX and Carlsbad was tiny, but they were able to manage it. I had removed the pedals, turned the handlebars sideways and attached them to the top tube, and let a bit of air out of the tires. It was a pretty easy ride from that tiny airport to the resort where I was staying, which was about a mile from the resort hosting the conference. As a bonus, one of the conference activities was a "mountain bike" ride on gravel trails and a small hill. The outfitters provided people with low end mountain bikes for the ride and were aghast that I would try the route on my Quickbeam. I assured them that the Quickbeam had been designed on paths far gnarlier than anything we would see, and I was right.
I've never done that again, having switched to a S&S coupled Bleriot with a hard and soft cases for travel. The soft case you can carry like a backpack, but it's awkward over any kind of distance. More recently I have been using an Orucase, which is too big to be manageable on the bike. You can fold it down to about the size of a small golf bag, but that's still awkward to carry.
I have no idea if airlines would still take an unboxed/unbagged bike, but it worked pretty well for me that one time. One argument in favor of that approach is the baggage handlers don't treat it like just another bag. I watched it getting loaded and unloaded and they were pretty careful with it. I think a single-speed/fixed-gear bike makes that approach way more feasible - way fewer things to get damaged.
Ted Durant
MIlwaukee, WI USA