How to get rid of visible edges when using the Curve modifier?

I’m trying to use the curve modifier to deform a cube along a bezier curve. Unfortunately I do get some nasty edges when rendering the results, even with smooth shading:

First I thought, I’d need more geometry, but even after adding a subsurf modifier the result is the same. There should be plenty of geometry available to bend the cube:

How can I get rid of those nasty edges from the first picture?

Increasing the resolution of the curve

Thanks a lot!!!

You are my hero, I was getting crazy already!

Have a nice day.