Can't speak for anyone else, but after doing some research I was told to pack my bike securely in a recycled bike box, which I then shipped cross country.
I watched them as they moved it from bus to bus and they handled it with kid gloves. They even placing it in a separate (tall) compartment upright all by itself. My bike arrived just as I had packed it. Not even the box was damaged. My experience with Greyhound was 5 star.