The nickelate Pr₄Ni₃O₈ features quasi-two-dimensional layers consisting of three stacked square-planar NiO₂ planes, in a similar way to the well-known cuprate superconductors. The mixed-valent nature of Ni and its metallic properties makes it a candidate for potentially unconventional superconductivity. We have synthesized Pr₄Ni₃O₈ by topotactic reduction of Pr₄Ni₃O₁₀ in 10% hydrogen gas, and report on measurements of powder-neutron diffraction, magnetization, and muon-spin rotation (µSR). We find that Pr₄Ni₃O₈ shows complicated spin-glass behavior with a distinct magnetic memory effect in the temperature range from 2 to 300 K and a freezing temperature T_s ≈ 68K. Moreover, the analysis of µSR spectra indicates two magnetic processes characterized by remarkably different relaxation rates: a slowly relaxing signal, resulting from paramagnetic fluctuations of Pr/Ni ions, and a fast-relaxing signal, whose relaxation rate increases substantially below ≈ 70K which can be ascribed to the presence of short-range correlated regions. We conclude that the complex spin-freezing process in Pr₄Ni₃O₈ is governed by these multiple magnetic interactions. It is possible that the complex magnetism in Pr₄Ni₃O₈ is detrimental to the occurrence of superconductivity.