Two-dimensional massive Dirac systems feature a gapped Dirac-cone dispersion central to a broad range of phenomena in condensed-matter and topological physics. While quantum simulations have demonstrated one-dimensional Dirac dynamics and two-dimensional massless Weyl dynamics, programmable simulation of massive two-dimensional Dirac dynamics remains experimentally unexplored. Here, we realize a programmable two-dimensional free Dirac Hamiltonian in circuit QED, with independently tunable spin-momentum couplings and mass, using a single Rabi-driven qubit coupled to two modes of a multimode cavity. Using this Hamiltonian, we observe rotational Zitterbewegung of a two-dimensional massive Dirac particle and its dependence on the effective mass. Time-dependent master-equation simulations incorporating measured decoherence, corrections beyond the rotating-wave approximation (RWA), and anharmonicity of the transmon reproduce the observed dynamics. Our results establish multimode circuit QED as a compact, programmable platform for higher-dimensional relativistic quantum dynamics and provide a foundation for exploring dynamical and topological phenomena in gapped Dirac systems.