Unidirectional droplet motion without energy input has attracted considerable attention in various potential applications. We report on the fabrication of unconventional laser-induced periodic surface structures (LIPSS) for unidirectional droplet micro-displacement by a femtosecond laser at a large incident angle. In order to overcome the mechanical durability of the existing soft materials such as PDMS for controlling the droplet, Zr-based bulk metallic glass (Zr-BMG) is used to fabricate the typical non-centrosymmetric unconventional LIPSS, which include micro-sized elliptical arc-shaped structures and nano-ripples with the central spatial periodicity of 400 nm. The spots and the grating of unconventional LIPSS on Zr-BMG are fabricated to construct the functional surface structures by femtosecond laser irradiation. The unidirectional micro-displacement of water droplet on Zr-BMG was achieved using functional surface structures. For better understanding the fabrication of unconventional LIPSS, the underlying formation mechanism was revealed by numerical simulations. This work gives a fast, precise and low-cost method to fabricate the non-centrosymmetric surface micro/nano-structures on metal materials for unidirectional droplet motion in microfluidics.