We have synthesized single crystals of Dirac semimetal candidates AZnBi2 with A=Ba and Sr. In contrast to A=Sr, the Ba material displays a local Zn vacancy ordering, which makes the observation of quantum oscillations in out-of-plane magnetic fields possible. As a Dirac semimetal candidate, BaZnBi2 exhibits a small cyclotron electron mass, high quantum mobility, and nontrivial Berry phases. Three Dirac dispersions are observed by angle-resolved photoemission spectroscopy and identified by first-principles band-structure calculations. Compared to AMn(Bi/Sb)2 systems which host Mn magnetic moments, BaZnBi2 acts as a nonmagnetic analog to investigate the intrinsic properties of Dirac fermions in this structure family.