Anonymization of event logs facilitates process mining while protecting sensitive information of process stakeholders. Existing techniques, however, focus on the privatization of the control-flow. Other process perspectives, such as roles, resources, and objects are neglected or subject to randomization, which breaks the dependencies between the perspectives. Hence, existing techniques are not suited for advanced process mining tasks, e.g., social network mining or predictive monitoring. To address this gap, we propose PMDG, a framework to ensure privacy for multi-perspective process mining through data generalization. It provides group-based privacy guarantees for an event log, while preserving the characteristic dependencies between the control-flow and further process perspectives. Unlike existin privatization techniques that rely on data suppression or noise insertion, PMDG adopts data generalization: a technique where the activities and attribute values referenced in events are generalized into more abstract ones, to obtain equivalence classes that are sufficiently large from a privacy point of view. We demonstrate empirically that PMDG outperforms state-of-the-art anonymization techniques, when mining handovers and predicting outcomes.