Dose-finding clinical trials in oncology aim to determine the maximum tolerated dose (MTD) of a new drug, generally defined by the proportion of patients with short-term dose-limiting toxicities (DLTs). Model-based approaches for such phase I oncology trials have been widely designed and are mostly restricted to the DLTs occurring during the first cycle of treatment, although patients continue to receive treatment for multiple cycles. We aim to estimate the probability of DLTs over sequences of treatment cycles via a Bayesian cumulative modeling approach, where the probability of DLT is modeled taking into account the cumulative effect of the administered drug and the DLT cycle of occurrence. We propose a design, called DICE (Dose-fInding CumulativE), for dose escalation and de-escalation according to previously observed toxicities, which aims at finding the MTD sequence (MTS). We performed an extensive simulation study comparing this approach to the time-to-event continual reassessment method (TITE-CRM) and a benchmark. In general, our approach achieved a better or comparable percentage of correct MTS selection. Moreover, we investigated the DICE prediction ability.