In this paper, we propose a novel high order unfitted finite element method on Cartesian meshes for solving the acoustic wave equation with discontinuous coefficients having complex interface geometry. The unfitted finite element method does not require any penalty to achieve optimal convergence. We also introduce a new explicit time discretization method for the ordinary differential equation (ODE) system resulting from the spatial discretization of the wave equation. The strong stability and optimal $hp$-version error estimates both in time and space are established. Numerical examples confirm our theoretical results.