We study spin excitation spectra of one-, two-, and three-dimensional magnets featuring nonmagnetic defects at a wide range of concentrations. Taking the Heisenberg model as the starting point, we tackle the problem by both direct numerical simulations in large supercells and using a semianalytic coherent-potential approximation. We consider the properties of the excitations in both direct and reciprocal spaces. In the limits of the concentration c of the magnetic atoms tending to 0 or 1 the properties of the spin excitations are similar in all three dimensions. In the case of a low concentration of magnetic atoms the spin excitation spectra are dominated by the modes confined in the real space to single atoms or small clusters and delocalized in the reciprocal space. In the limit of c tending to 1, we obtain the spin-wave excitations delocalized in the real space and localized in the reciprocal space. However, for the intermediate concentrations the properties of the spin excitations are strongly dimensionality dependent. We pay particular attention to the formation, with increase of c, of the Lorentzian-shaped peaks in the spectral densities of the spin excitations, which can be regarded as magnon states with a finite lifetime given by the width of the peaks. In general, low-dimensional magnets are more strongly affected by the presence of nonmagnetic impurities than their bulk counterparts. The details of the electronic structure, varying with the dimensionality and the concentration, substantially influence the spin excitation spectra of real materials, as we show in the example of the FeAl alloy.