In this paper, we introduce a class of $(P, ω)$-partitions that we call periodic $(P, ω)$-partitions, then prove that such $(P, ω)$-partitions satisfy a homogeneous first-order matrix difference equation. After defining an appropriate counting problem for the above $(P, ω)$-partitions, we show that as a consequence of this equation, periodic $(P, ω)$-partitions can be enumerated with constant coefficient linear recurrence relations. By analysing the above matrix difference equation, we also prove a result for the asymptotic growth rate for the number of periodic $(P, ω)$-partitions. The results of this paper generalizes and strengthens the constant coefficient linear recurrence results proved by Sun and by López, Martínez, Pérez, Pérez, and Basova for enumerating standard Young tableaux on shifted strips with constant width.