Process mining is a family of techniques that aim at analyzing business process execution data recorded in event logs. Conformance checking is a branch of this discipline embracing approaches for verifying whether the behavior of a process, as recorded in a log, is in line with some expected behavior provided in the form of a process model. Recently, techniques for conformance checking based on declarative specifications have been developed. Such specifications are suitable to describe processes characterized by high variability. However, an open challenge in the context of conformance checking with declarative models is the capability of supporting multi-perspective specifications. This means that declarative models used for conformance checking should not only describe the process behavior from the control flow point of view, but also from other perspectives like data or time. In this paper, we close this gap by presenting an approach for conformance checking based on MP-Declare, a multi-perspective version of the declarative process modeling language Declare. The approach has been implemented in the process mining tool ProM and has been experimented using artificial and real-life event logs.