One of the most promising strategies to treat cancer is attacking it with viruses designed to exploit specific altered pathways. Here, the effects of oncolytic virotherapy on tumors having compact, papillary, and disconnected morphologies are investigated through computer simulations of a multiscale model coupling macroscopic reaction-diffusion equations for the nutrients with microscopic stochastic rules for the actions of individual cells and viruses. The interaction among viruses and tumor cells involves cell infection, intracellular virus replication, and the release of new viruses in the tissue after cell lysis. The evolution over time of both the viral load and cancer cell population, as well as the probabilities for tumor eradication, were evaluated for a range of multiplicities of infection, viral entries, and burst sizes. It was found that in immunosuppressed hosts, the antitumor efficacy of a virus is primarily determined by its entry efficiency, its replicative capacity within the tumor, and its ability to spread over the tissue. However, the optimal traits for oncolytic viruses depend critically on the tumor growth dynamics and do not necessarily include rapid replication, cytolysis, or spreading, currently assumed as necessary conditions for a successful therapeutic outcome. Our findings have potential implications on the design of new vectors for the viral therapy of cancer.