Multiparticle collision dynamics (MPCD) is a mesoscale particle-based simulation method for simulating hydrodynamic interactions in soft materials. MPCD uses a particle-based solvent that exchanges momentum through spatially localized stochastic collisions. Colloidal particles can be represented in MPCD by discretizing their surfaces into particles that also participate in these collisions, and this approach has recently been shown to reliably describe the diffusion and sedimentation of colloidal particles with a variety of shapes. However, in this prior work, a network of stiff springs was needed to maintain the discretized model for the colloidal particles, creating a tedious process for configuring the simulations and restricting the integration timestep. In this work, we overcome these limitations by incorporating rigid-body constraints into MPCD. We implement these constraints in the HOOMD-blue simulation package with support for CPUs, GPUs, and domain decomposition, then characterize the accuracy and performance of our rigid-body approach compared to stiff springs. Our work enables convenient MPCD simulations of suspensions of colloidal particles with complex shapes.