Depending on the airline, bringing a bicycle along in an official box can cost several hundred dollars. State Bicycle Co. breaks down the "Bananas Box & Chop Method", which drops the cost to a normal checked bag fee.
You'll need to start with a bike box (usually a bike shop has these around), then cut the box down to a 75cm x 75cm square, stuff everything inside, then chop the corner to get it within the designated size of 140 linear centimetres. It's obviously a bit more complicated than randomly tossing parts into a bike box, but it's a heck of a lot cheaper. Of course, not all airlines charge the extra fee (and they're not all several hundred dollars), so be sure to check in with your airline before you go through the trouble of hacking together your own box.