The emission of pure spin currents by mechanical deformations, the piezospintronic effect, in antiferromagnets is studied. We characterize the piezospintronic effect in an antiferromagnetic honeycomb monolayer in response to external strains. It is shown that the strain tensor components can be evaluated in terms of the spin Berry phase. In addition, we propose an experimental setup to detect the piezospin current generated in the piezospintronic material through the inverse spin Hall effect. Our results apply to a wide family of two-dimensional antiferromagnetic materials without inversion symmetry, such as the transition-metal chalcogenophosphate materials MPX3 (M=V, Mn; X=S, Se, Te) and NiPSe3.