Since I've started my training in Muay Thai, I've come to realize just how important it is to have a flexible back. What do I do to improve flexibility? Back bridges? What else? I try to lay on my stomach, keep my quads and hips on the ground and push my chest as far back as I can, but it doesn't seem to do much. Any help is appreciated.

I think what he's describing is the cobra stretch.

Anyway, I'm not quite sure what you mean by 'back flexibility'. A good backbend is dependent on the flexibility of the muscles in the front of your torso - they're the ones being stretched. Good flexibility in the muscles of the back gives a good frontbend.

Anyway some stretches for backbending:
Back bridge. A good bridge should look like this http://www.gsis.edu.hk/aactivities/Images/gym_bridge.JPG. Note how the legs are nearly straight.
Cobra stretch. Preferably put your hands on a chair or something for elevation
Backhold.
After backbending, do this counterstretch to prevent compression or tension in the lower back.

For frontbending, try to get your legs behind your head. Or stretch down like you're going to touch your toes, and pull your head through between your legs.

Thanks guys. When we throw our knees or front kicks, we arch our backs for additional power and reach. My back always hurts after throwing a lot of knees. I really want to work on it.

That would be more about strenthening your back than making it more flexibile.