While dynamic policies have historically formed the foundation of most influential papers dedicated to the joint replenishment problem, we are still facing profound gaps in our structural understanding of optimal such policies as well as in their surrounding computational questions. To date, the seminal work of Roundy (1985, 1986) and Jackson et al. (1985) remains unsurpassed in efficiently developing provably-good dynamic policies in this context. The principal contribution of this paper consists in developing a wide range of algorithmic ideas and analytical insights around the continuous-time joint replenishment problem, culminating in a deterministic framework for efficiently approximating optimal dynamic policies to any desired level of accuracy. These advances enable us to derive a compactly-encoded replenishment policy whose long-run average cost is within factor $1 + ε$ of the dynamic optimum, arriving at an efficient polynomial-time approximation scheme (EPTAS). Technically speaking, our approach hinges on affirmative resolutions to two fundamental open questions: -- We devise the first efficient discretization-based framework for approximating the joint replenishment problem. Specifically, we prove that every continuous-time infinite-horizon instance can be reduced to a corresponding discrete-time $O( \frac{ n^3 }{ ε^6 } )$-period instance, while incurring a multiplicative optimality loss of at most $1 + ε$. -- Motivated by this relation, we substantially improve on the $O( 2^{2^{O(1/ε)}} \cdot (nT)^{ O(1) } )$-time approximation scheme of Nonner and Sviridenko (2013) for the discrete-time joint replenishment problem. Beyond an exponential improvement in running time, we demonstrate that randomization and hierarchical decompositions can be entirely avoided, while concurrently offering a relatively simple analysis.