It is perfectly feasible to take your bikes on most airlines, the rules and regulations do vary from carrier to carrier. Most airlines require you now to bag or box your bike, very few just allow you to remove the pedals, deflate tyres and turn the handle bars around. You will need to check with the airline that you intend to use well before booking as if you turn up with an un-bagged bike the airline may refuse to handle it!. Most of the airlines include the weight of you bike in you luggage allowance, so be warned you could incur excess baggage charges!!
Our first experience of taking our bikes on planes was on our 2000 tour of Finland when we flew with Buzz from Stansted to Helsinki. Their regulations then were that the bikes should have both wheels and pedals removed, tyres deflated and handle bars turned around. We decided that as they required both wheels removed we would pack the bikes in bike bags.
We purchased two bike bags from St Johns cycles in Bridgewater as they seemed the biggest in size to cope with our large touring bikes and the best value. We then had the problem of how to get to the airport, we decided on the outward to go by train as Stansted has it's own terminus.
This meant that we could pack the bike bags at home saving the hassle of having to take them to pieces in the airport concourse (this was the right decision as Stansted was heaving with people that day) We found the bags reasonably easy to carry although it helps to be tall! On our return journey we were picked up at the airport by mother in law with our car and the bike rack. This did involve rebuilding the bikes in the car park but then there was not the pressure of meeting flight departure times.
Although our bikes did suffer some minor damage on the outward journey (a broken bottle cage and a crack in the plastic chain guard), we think this was mainly due to them being opened by customs in order to have their wheels sprayed against 'Foot & Mouth' and then not being repacked properly.
Our second flight with the bikes was with Norwegian Airlines to Norway in 2006, again we used the bike bags and had no real problems, although the bike bags did suffered a bit of scuffing and rips but the bikes were un-damaged.