Sorry for the delayed response. Was visiting relatives and caught the flu.
If you are using meshes, there may be some answers in the forums dealing with modeling. That is an interesting pattern as it appears that the moire is appearing in areas where there are gaps in the mesh. Almost as if is is created by light reflections on top of the slats.
Would smoothing help? I assume that each slat is a single plane. If so, it might help to use 2 planes, including a small plane at the top which has a bit of a bevel (like Blender Guru did to smooth the top of his coffee cup).
Yes, I did mean a normal map (the kind with the purple background). But that was before I realized that the surface you have created is transparent. In that case, an opacity map does make more sense. Or you could interleave transparent planes between the slats or put a single transparent plane behind (or in front of) the mesh. You could vary the properties of the plane, to prevent reflections or to make it only mostly transparent.
These are just ideas and I am not sure if any will work. Unfortunately, when a program displays a series of lines that are closely parallel with each other, there is a tendency for a moire pattern to appear when the lines are tilted. But I am not 100% that is what is causing the problem here.