The Lattice Boltzmann Method (LBM), e.g. in [6, 10], can be interpreted as a partial differential equation solver. Although the LBM is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical tool, allows it to be extended to solid mechanics as well. In this work, an existing LBM for the two-dimensional wave equation is extended to more general plane strain problems. The proposed algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively. A particular focus is on the implementation of boundary conditions that are commonly encountered in applied solid mechanics: Dirichlet and Neumann boundary conditions. Last, a numerical experiment is conducted that highlights the feasibility of the new LBM in comparison to the Finite Element Method.