Generally a horse rushes b/c he's either 1) afraid of something 2) in pain.
Dressage (good flatwork, whatever you want to call it) is crucial just as MIE said, but there could be a whole lot more to it then just dressage, esp if he's green.
Is he rushing IN the line? Horses that rush, esp the 2nd jump of the line, often do it b/c you come in too slowly to the first one and it becomes uncomfortable to jump the 2nd, so they respond by running in the line. This actually can stem from coming out of the corner at too slow a pace. People's tendancy on a horse that rushes is to slow them down, but this usually is NOT the answer. The answer is to find the "optimal" speed and stay with it. Dressage comes in handy so that when you tell your horse to slow down or stay on the same pace, they actually listen!
If your trainer is telling you to let him figure it out I'm assuming he's a little green and I wonder if he's just not sure what to do with himself. The hardest thing for a horse when learning how to jump is what to do with their body. This is where gymnastics are key! Gymnastics, gymnastic, and more gymnastics. They will teach him where to leave from the ground, where to place his legs, how to hold his body, etc etc. When they don't know this stuff they worry and some horses react by running. Lots of bounces and series of gymnastics are a great way to make a horse back off on their own and think through the jumps. This would be my biggest recommendation to you.
Alsol, it is very important that you hold your body still, relaxed, balanced over the fence. You want to be secure so you don't hit him in the mouth over the fence or hit him in the back before/over/after the fence, or duck over the fence. Be sure you don't hang on his mouth or clench your body in any way. Any hot horse (and some not so hot) will react by running.
Last but not least, make sure there isn't something hurting him! Soundness, saddle fit, etc.