A novel EigenWave algorithm is described to compute eigenvalues and eigenfunctions of elliptic boundary value problems. Based on the recently developed WaveHoltz scheme, the algorithm solves a related time-dependent wave equation as part of an iteration. At each iteration, the solution is filtered in time. After filtering, the solution mainly contains eigenmodes whose eigenvalues are near the target frequency of the filter. The iteration is embedded within a matrix-free Arnoldi algorithm, allowing the efficient computation of multiple eigenpairs near the target frequency. The approach allows the computation of eigenvalues anywhere in the spectrum without the need to invert an indefinite matrix, as is common with other approaches. The approach is demonstrated by finding eigenpairs of the Laplacian in complex geometry using overset grids. Results in two and three-space dimensions are presented. For large enough problems, it is demonstrated that EigenWave can outperform modern Arnoldi-type eigenvalue algorithms.