In this article we prove for $1<p<\infty $ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega $. More precisely, $\Omega $ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega $ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as a Fourier multiplier operator.