The moiré pattern observed in stacked noncommensurate crystal lattices, such as heterobilayers of transition metal dichalcogenides, produces a periodic modulation of their band gap. Excitons subjected to this potential landscape exhibit a band structure that gives rise to a quasiparticle dubbed the moiré exciton. In the case of MoS_{2}/WSe_{2} heterobilayers, the moiré trapping potential has honeycomb symmetry and, consequently, the moiré exciton band structure is the same as that of a Dirac-Weyl fermion, whose mass can be further tuned down to zero with a perpendicularly applied field. Here we show that, analogously to other Dirac-like particles, the moiré exciton exhibits a trembling motion, also known as Zitterbewegung, whose long timescales are compatible with current experimental techniques for exciton dynamics. This promotes the study of the dynamics of moiré excitons in van der Waals heterostructures as an advantageous solid-state platform to probe Zitterbewegung, broadly tunable by gating and interlayer twist angle.