The dynamics of passive Lagrangian tracers in three-dimensional quasigeostrophic turbulence is studied numerically and compared with the behavior of two-dimensional barotropic turbulence. Despite the different Eulerian properties of the two flows, the Lagrangian dynamics of passively advected tracers in three-dimensional quasigeostrophic turbulence is very similar to that of barotropic turbulence. In both systems, coherent vortices play a major role in determining the mixing and dispersion properties. This work indicates that recent results on particle dynamics in barotropic, two-dimensional turbulence carry over to more realistic baroclinic flows, such as those encountered in the large-scale dynamics of the atmosphere and ocean.