If I play with the pivot points of the balls AND zero out the rings so that they are flat on each plane, I can get the ball to follow the shape of the ring by using roll or yaw or whatever depending on the plane I am in.

The problem is that once the rings are not on a flat plane and are setup at an angle like shown, the balls will no longer follow the ring.

I have even tried grouping them and then just rotating the ring, but the rings won't rotate around the sphere properly at that angle either.

Any ideas what I can do?

Maybe a job for a Python script?
Maybe add the balls to the side of an invisible cylinder and tilt the cylinder to the plane of the visible rings?

"The problem is that once the rings are not on a flat plane and are setup at an angle like shown, the balls will no longer follow the ring."
Maybe make everything horizontal and vertical, if that keeps stuff aligned (except the center disc/sphere) and tilt only the camera? 

OK, I decided to try your advice and just make the two planes vertical and horizontal. It is not ideal because the planes now have to be perpendicular to each and the angles aren't quite what I want anymore, but the orbit thing works.

The cylinder thing had the same results of my first attempt, it rotates around the wrong angle.

If anyone knows of a way to have the rotation/orbit work on different angles let me know, but I am going to try it with the perpendicular angles for now.

Glad it more or less works. Hope we get to see it in action.

here it is