The Replenishment Storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system where each item has a given reorder size and cycle length. The reorders can only take place at integer time units within the cycle. This problem was shown to be weakly NP-hard for constant joint cycle length (the least common multiple of the lengths of all individual cycles). When all items have the same constant cycle length, there exists a Fully Polynomial Time Approximation Scheme (FPTAS), but no FPTAS has been known for the case when the individual cycles are different. Here we devise the first known FPTAS for the RSP with different individual cycles and constant joint cycle length.