Solar coronal mass ejections (CMEs) are associated with many dynamical phenomena, among which EIT waves have always been a puzzle. In this Letter MHD processes of CME-induced wave phenomena are numerically simulated. It is shown that as the flux rope rises, a piston-driven shock is formed along the envelope of the expanding CME, which sweeps the solar surface as it propagates. We propose that the legs of the shock produce Moreton waves. Simultaneously, a slower moving wavelike structure, with an enhanced plasma region ahead, is discerned, which we propose corresponds to the observed EIT waves. The mechanism for EIT waves is therefore suggested, and their relation with Moreton waves and radio bursts is discussed.