Strong-field perturbation calculations are performed of the gravitational radiation emitted when spheroidal dust shells of mass m and arbitrary eccentricity fall radially into Schwarzschild black holes of mass M much greater than m. The scaling laws obeyed by the numerical results in limiting eccentricity regimes are derived analytically. It is demonstrated that the total energy emitted in gravitational radiation by a nonspherical dust cloud falling into a black hole is always less than the energy radiated by a point particle of the same mass falling into the hole. The suppression results from interference between waves emitted from different regions of the extended, infalling mass. Nonspherical, radial dust calculations suggest that efficient gravitational wave generation from collapse requires either matter pressure sufficiently strong to induce bounces and shocks or angular momentum. Both the numerical and analytic results should prove useful for the testing of general relativistic, 2+1 dimensional (axisymmetric) hydrodynamical codes which treat both matter and fields.