Greyhound is really the way to go if it's a trip of under, say, 400 miles. But you'll need to work the system to the most out of it.
Yes, some of the other riders can a tad scary. But overall, it's as safe as air travel.
Your bike will have to be boxed of course, just like most other forms of travel. Stay out of the big city bus stations and you'll be fine. Remember the bus stops at little stops all over, making it slow. But it also means you can catch it more places. If you pick it up at a small town stop (even if it's 10 miles away from a dirty, big city bus depot), you'll make sure your bike is on the same bus as you, and the question of paying extra for you bike will most likely never be asked.
Also try to have the bus drop you off at a small, outside the city, stop. riding out of a metro bus station (or airport) is often stressful.