The circular dichroism (CD) of a material is the difference in optical absorption under left- and right-circularly polarized illumination. It is crucial for a number of applications, from molecular sensing to the design of circularly polarized thermal light sources. The CD in natural materials is typically weak, leading to the exploitation of artificial chiral materials. Layered chiral woodpile structures are well known to boost chiro-optical effects when realized as a photonic crystal or an optical metamaterial. We here demonstrate that light scattering at a chiral plasmonic woodpile, which is structured on the order of the wavelength of the light, can be well understood by considering the fundamental evanescent Floquet states within the structure. In particular, we report a broadband circular polarization bandgap in the complex band structure of various plasmonic woodpiles that spans the optical transparency window of the atmosphere between 3 and 4 μ m and leads to an average CD of up to 90% within this spectral range. Our findings could pave the way for an ultra-broadband circularly polarized thermal source.