It used to be BikeFlights but they've just switched to UPS so the prices have gone up. Your best bet will probably be FedEx, although I've heard ShipBikes is pretty good? I haven't used them yet though. Some bike shops will save you a bike box, these are nice because they're usually designed to be within the acceptable dimensions.
I used to ship car parts greyhound, it's pretty funny. I have some muscle bikes I'd like to sell, I just don't feel like dealing with the shipping , I 've never shipped a bike myself.
Bikeflights was indeed the cheapest way. As mentioned, since their switch to ups, prices have gone up. From what I was told, the prices are supposed to come back down a little once the new partnership ‘settles’ but I wouldn’t hold my breath.
I recently shipped a bike via ups from California to North Carolina, it was about $100.