We consider an atomistic model to investigate the flux of water through thin Linde type A (LTA) zeolite membranes with differing surface chemistries. Using molecular dynamics, we have studied the flow of water under hydrostatic pressure through a fully hydrated LTA zeolite film (similar to 2.5 nm thick) capped with hydrophilic and hydrophobic moieties. Pressure drops in the 50-400 MPa range were applied across the membrane, and the flux of water was monitored for at least 15 ns of simulation time. For hydrophilic membranes, water molecules adsorb at the zeolite surface, creating a highly structured fluid layer. For hydrophobic membranes, a depletion of water molecules occurs near the water/zeolite interface. For both types of membranes, the water structure is independent of the pressure drop established in the system and the flux through the membranes is lower than that observed for the bulk zeolitic material; the latter allows an estimation of surface barrier effects to pressure-driven water transport. Mechanistically, it is observed that (i) bottlenecks form at the windows of the zeolite structure, preventing the free flow of water through the porous membrane, (ii) water molecules do not move through a cage in a single-file fashion but rather exhibit a broad range of residence times and pronounced mixing, and (iii) a periodic buildup of a pressure difference between inlet and outlet cages takes place which leads to the preferential flow of water molecules toward the low-pressure cages.