A prerequisite for the realization of solid-state batteries is the development of highly conductive solid electrolytes. Li3PS4 is the archetypal member of the highly promising thiophosphate family of Li-ion conductors. Despite a multitude of investigations into this material, the underlying atomic-scale features governing the roles of and the relationships between cation and anion dynamics, in its various temperature-dependent polymorphs, are yet to be fully resolved. On this basis, we provide a comprehensive molecular dynamics study to probe the fundamental mechanisms underpinning fast Li-ion diffusion in this important solid electrolyte material. We first determine the Li-ion diffusion coefficients and corresponding activation energies in the temperature-dependent γ, β, and α polymorphs of Li3PS4 and relate them to the structural and chemical characteristics of each polymorph. The roles that both cation correlation and anion libration play in enhancing the Li-ion dynamics in Li3PS4 are then isolated and revealed. For γ- and β-Li3PS4, our simulations confirm that the interatomic Li-Li interaction is pivotal in determining (and restricting) their Li-ion diffusion. For α-Li3PS4, we quantify the significant role of Li-Li correlation and anion dynamics in dominating Li-ion transport in this polymorph for the first time. The fundamental understanding and analysis presented herein is expected to be highly applicable to other solid electrolytes where the interplay between cation and anion dynamics is crucial to enhancing ion transport.