In a dense discrete random medium, the propagation and scattering of waves are affected not only by the individual properties of the particles such as size, shape, and permittivity, but also by group properties such as the statistics of relative particle positions and relative orientations. We use Monte Carlo simulations to investigate the interactions of electromagnetic waves with a dense medium consisting of spheroidal particles for cases of random orientation and for cases of aligned orientation. A shuffling process is used to generate the positions of densely packed spheroids. Multiple-scattering equations are formulated by means of the volume integral equation and are solved numerically. The scattering results are averaged over realizations. Numerical results are presented for the extinction rates and the phase matrices. Salient features of the numerical results indicate that (1) the extinction rates of densely packed small spheroids are smaller than those of independent scattering; (2) for aligned spheroids, the extinction rates are polarization dependent; and (3) the co-polarized part of the phase matrix for densely packed spheroids is smaller than that of independent scatering, while the cross-polarized part is larger than that for independent scattering. This means that the ratio of cross-polarization to co-polarization is significantly higher than that of independent scattering.