My first guess would be a bad engine or transmission mount. The drive shaft should vibrate whether you're on the gas or not, but a broken mount could cause the car to shake in neutral because you've got no engine load (torsion) on it.
Does it shake while stopped neutral as well, or only while moving? What kind of shape is your balancer in? My next guess would be the rubber in the stock balancer. They tend to rot out over the years and I've seen them turn, even fly right off before. That'll throw things out of balance in a hurry, although you would normally feel it in gear as well.