We present a novel architecture for very thin Frequency Selective Surfaces (FSS) and waveguide passband filters based on resonantly transparent super-thin printed screens, transversely placed on the waveguide axis. The screens are based on stacked metal laminate layers where each layer is the negative print of each adjacent layer along the propagation axis. Hence, these filters are composed of stacked layers obeying the Babinet duality. We provide two optimized designs of such filters, operating both as Frequency-Selective Surfaces and as finite screens placed transversely inside a rectangular waveguide. We show that these optimized designs have very small thicknesses, relative to their performance. Therefore, they are an attractive candidate for replacing traditional bulky and heavy waveguide filters based on sparsely placed metallic posts along a substantial length of waveguide sections.