Neons are sensitive to wheel balance. If it is not perfect, it induces a resonance that causes steering wheel shimmy. It is generally referred to as the 73mph shake. At the factory, Neon wheel and tire assemblies are balanced within 1/8 of an ounce. Most other tire stores and such can only balance within 1/4 of an ounce and that's assuming that their wheel balancer is calibrated properly (rare) and the operator knows how to run the machine. The resonance occurs due to the front lower control arm rear bushing. When this bushing gets worn, the shake certainly gets worse.

To eliminate this problem, make sure your wheels are really balanced. Next, replace the front lower control arm rear bushings (an arbor press is required). If you want to reduce the sensitivity to wheel balance and move the resonant frequency (speed at which shimmy happens) up, install some stiffer bushings like the polyurethane Energy Suspension pieces in place of the factory ones. They will make the car ride more harshly over small bumps because the wheel will not recess reward, but it's a worthy compromise.