I've traveled with both a "normal" diamond-framed bike and with my Bike Friday. The Bike Friday is made to travel, and folds into a normal-sized suitcase so there aren't any oversize baggage charges. But that's not what you're asking...

I've traveled with a diamond-framed bike packed in a box, and with it packed in a soft-sided bike case.

On my first trip with it, I was just a neophyte at bike assembly. I made an appointment with a mechanic at my LBS so I could learn how to pack it. They supplied a box, and I packed it with someone giving me direction. I shipped it UPS to my destination, and I chose the lazy gals way out. I shipped it to a bike shop, and they put it back together for me. On my return, I went to another bike shop (it was a one-way trip). They gave me a box and loaned me full-sized tools and a workstand. I packed the bike myself, and I checked it as baggage on the plane. And yes, I did have to pay oversize baggage fees - which were more than the UPS fees.

Later I bought a soft-sided case (yes, I know, hard-sided is better, but I needed to be able to squish the thing up and ship it to myself for the end of my trip). I liked using the case better than the cardboard box, but it did require paying oversize baggage fees in both directions.

I haven't flown with my diamond-framed bike since I got a Bike Friday for travel - but the Bike Friday doesn't get rid of the security issues. When I flew home from Hawaii with the bike, the airport in Kona required that all bags be opened and inspected. I watched as the TSA guy opened the bike case and started removing pieces. Yikes! I was allowed to give him hints on how to put it back - but I wasn't allowed to touch anything. Still, everything worked out OK.

If you want to use a bike case, maybe your LBS has some available for rent. Or maybe a local club rents one or knows someone who is willing to loan their case out.