Periodically patterned thin films of slanted silver nanocolumns were deposited by directing a collimated vapor flux of silver toward square and hexagonal gratings of photoresist on glass substrates. Angle-resolved specular-transmittance measurements in the visible and near-infrared wavelength bands on these periodically patterned columnar thin films (CTFs) were carried out to investigate the excitation of surface plasmon-polariton (SPP) waves bound tightly to either the air/CTF or the photoresist/CTF interfaces. The orientation of the propagation vector of the incident p-polarized plane wave with respect to the morphologically significant plane of the CTFs was varied to reveal asymmetric (unidirectional) coupling of Floquet modes to SPP waves. The asymmetric coupling is maximal when the propagation vector of the incident plane wave lies wholly in the morphologically significant plane. Theoretical understanding based on the Bruggeman formalism to homogenize the silver CTFs into hyperbolic biaxial continua is able to explain the experimental observations very well.