We have studied the static and dynamic properties of quasi-two-dimensional (2D) quantum antiferromagnets (AFs) diluted with spinless impurities using spin-wave theory and T-matrix approximation. We show that the spectrum of a 2D AF at long wavelengths is overdamped at an arbitrary concentration of spinless impurities. The scattering leads to a length scale l/a∼e π/4x, x being impurity concentration and a the lattice spacing, beyond which the influence of impurities on the spectrum is dominant. Although the dynamical properties are significantly modified we show that 2D is not the lower critical dimension for this problem. Thus, in low-dimensional systems with disorder the connection between static and dynamic quantities is not straightforward. Our results are in quantitative agreement with the recent Monte Carlo simulations and experimental data for S=1/2, S=1, and S=5/2. We have also proposed experiments which can further test the results of our theory.