Replying to an older post here, but thought I would reply since I just flew cross country with my GCC and with a little bit of work and the right equipment it was a super easy trip.
There are officially only 2 major US airlines that allow birds in the cabin- Air Alaska and Delta. We used Delta and you have to call in to make a pet reservation. Make sure you have specific dimensions and weight of your carrier so they can process the ticket correctly.
Like others have said on the Forums- the Celltei airline approved travel carrier is a GREAT investment. It measures 16" long, 10" wide and 11/5" tall, and that height can go down to 8" since it compresses to fit under the seat. Get your bird used to the carrier, have them eat it in it, take them for a few car rides (with the carrier on the floor) and just walk them around and let them understand they will be moving around in this. Our GCC did not love getting in, but I can grab her fairly easy and once in it, and a few treats, she was very content in there.
Delta says they have a special service counter for pet check in, but be aware that there is no special counter (at least in O'Hare) you have to stand in line for a regular service agent, so get to the airport a bit earlier than usual. You will pay your PET FEE at the service counter and they will issue you a ticket- currently to travel with a bird in cabin it is a $95 charge. Also realize that the carrier counts as a personal carry on item, so if you have a backpack or handbag as well- you would have to check any luggage with you.
TSA- just be prepared that you will have to take your bird out. I have seen some posts where they say agents will just look in- but at least for the Celltei carrier since it has only 3 sides that you can see into - I had numerous supervisors tell me that they have to run it through the machine to inspect the floor and the ceiling of the carrier. A TSA agent will take the carrier, while you walk through the detector, and then they escort you to a private room. I tried to just grab her out of the carrier but of course she wanted to explore the room! TSA agents were very friendly and thought it was a unique experience. Once again since she lets me grab her easily, I was able to get her back in when they returned the cage.
I kept the carrier fairly sparce, just millet on the floor, some treats scattered, and then in one of the food cups I placed some pieces of tangerine and watermelon. The fruit was a way for her to stay hydrated without water sloshing all over- although I did give her some sips of water on our layover by putting a cup in the carrier.
If you are ok on paying up a little- the Delta comfort seats have a bit more room underneath and allow you to fit the carrier without much effort under the seat in front of you. I would recommend an aisle seat just so you have some extra leg room if needed.
We are lucky to have a fairly quiet GCC so she thought the whole thing was a fun field trip. She only squeaked a few times and with the airplane noise you could barely hear it. There was a crying baby 2 rows in back of us and my seat mate turned to me during the flight and said "I'm much happier sitting next to the bird."
Good luck to any of those traveling in the future- hope more airlines realize what good travelers are feathered friends can be.