I've commuted to the local airport in La Crosse several times without any problem. (I also rode once to the Orange County airport, though never for a flight, and often fantasized about riding to the Denver airport.)

Prior to the first time I tried this, I took an exploratory ride to the airport to look for road hazards, possible delays, and bike parking. Everything went well. Then, I called the airport to make sure that it would be okay to park a bike there for several days.

On my airport trips, I ride my backup bike (a total monstrosity) to the airport, and throw a couple of locks on it. I also lock up a saddle bag containing any tools I do not want to take on the plane. Since I tend to pack light and don't care for roller bags, I take a large messenger bag. Compared to a backpack, I think it's easier to flatten out clothes so they don't get too wrinkled.

The only minor trouble I ran into was the time I forgot to take a 15mm hex wrench out of my bag. TSA measured it and said it was cool, so on the way back, I made sure to put it in the security bin along with my toiletries, laptop, and other such items. They seemed to appreciate that.

For what it's worth, the ability to ride to the airport has made my life a lot easier. We have no bus service to the airport, and cab service in this town is notoriously unreliable - as in they don't show up, arrive hours late, and so on. There is no such thing as a "cab stand" in La Crosse, even at the airport. Since we got rid of the household car, my only other option is to beg somebody for a ride - not really my style.