  The \emph{Product Structure Theorem} for planar graphs (Dujmovi\'c et al.\
\emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in
the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a
simple linear-time algorithm for finding this decomposition as well as several
related decompositions. This improves on the previous $O(n\log n)$ time
algorithm (Morin.\ \emph{Algorithmica}, \textbf{85}(5):1544--1558).
