I had the rims and tires replaced on my '97 Accord several months ago. American Racing Rims and Bridgestone Terazzo(?) tires. I now have a shimmy when my speed is 50mph +. I had the front end checked right away because it was aligned and balanced when they were put on. The recheck showed that it was fine. I had the tires balanced again when I had them rotated hoping that would solve it. No go. The shimmy is not terrible, just annoying.

What can be done to stop it?

take it to a real alignment shop.

12.3 What's a hubcentric wheel? Why do I need hubcentric adapters?

OEM wheels are hubcentric. They are produced in such a way that they fit snugly
on the hub (wheel mount location). The hub has a ridge on it that mates to the
backside of the wheel. This is done to ensure proper centering
of the wheel on the hub.

Aftermarket wheels are usually overbored on the backside for different car applications.
Special adapter rings are manufactured for individual cars to ensure their wheel is
hub-centered when mounted.

If not properly centered, you can have a steering wheel shimmy.

if all else fails, it could be a bent axle, our chrylser minvan had that problem when they took out our tranny to put new seals on it, they bent the axle slightly and it caused it to shake a little while we were goin down the road. we got numerous alignments and balancing before that and it didn't fix it.

I would have to agree with Rice-Rocketeer on the hub centric rings.... I have had the same issue my self...

American racing is notorious for the problems mentioned, sounds like you may need spacers.
Even then it may not be a perfect fit.
Is ther any vibratioon sound when you are driving or any clicking when you turn comming from you front end?

I will investigate the hub rings. I know that the original A/R rims had them. I don't know if the shop put on new ones for these. I don't have any noise associated with the shimmy other than the windshield now has an annoying matching sqeak.

A different alignment shop idea was going to be my next step.

I was looking for some kind of common thread and it looks like it is the A/R rims.

Thanks