I've imported some rings and a logo from illustrator and made them all 3D layers. I'm rotating the rings over time using simple expressions. Problem is when a ring rotates to the point where the camera is looking at the edge of the flat 2D layer. During that frame, it's almost like the bounding box of the layer becomes visible.
Here's the video. You can see what I'm talking about at the 6 second mark.
I can't see any bounding box- it's just a flat layer that is viewed from the side. You can try to add motion blur- that may solve your issue, or build extruded 3d rings based on the ring layers. A trick way to give depth that will also increase render time is to create multiple instances of the same layer parented to one that keeps the initial animation, and then move them one unit above or below. This, together with motion blur will give you "thickness" to your circles.