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We show that if $\mathbb Z^3$ can be tiled by translated copies of a set$F\subseteq\mathbb Z^3$ of cardinality the square of a prime then there is aweakly periodic $F$-tiling of $\mathbb Z^3$, that is, there is a tiling $T$ of$\mathbb Z^3$ by translates of $F$ such that $T$ can be partitioned intofinitely many $1$-periodic sets.

A Periodicity Result for Tilings of $\mathbb Z^3$ by Clusters of Prime-Squared Cardinality