Measurements in the heliosphere and high-resolution fluid simulations give clear indications for the anisotropy of plasma turbulence in the presence of magnetic fields. How this anisotropy affects transport processes like diffusion and dispersion remains an open question. The first efforts to characterize Lagrangian single-particle diffusion and two-particle dispersion in incompressible magnetohydrodynamic (MHD) turbulence were performed a decade ago. We revisit those pioneering results through updated simulations performed at higher Reynolds number. We present new investigations that use the dispersion of many Lagrangian tracer particles to examine the extremes of dispersion and the anisotropy in direct numerical simulations. We then point out directions in which Lagrangian statistics need to be developed to address the fundamental problem of anisotropic MHD turbulence and transport in solar and stellar winds.