Kinesin-1 ensembles maneuver vesicular cargoes through intersections in the 3-dimensional (3D) intracellular microtubule (MT) network. To characterize directional outcomes (straight, turn, terminate) at MT intersections, we challenge 350 nm fluid-like liposomes transported by ~10 constitutively active, truncated kinesin-1 KIF5B (K543) with perpendicular 2-dimensional (2D) and 3D intersections in vitro. Liposomes frequently pause at 2D and 3D intersections (~2s), suggesting that motor teams can simultaneously engage each MT and undergo a tug-of-war. Once resolved, the directional outcomes at 2D MT intersections have a straight to turn ratio of 1.1; whereas at 3D MT intersections, liposomes more frequently go straight (straight to turn ratio of 1.8), highlighting that spatial relationships at intersections bias directional outcomes. Using 3D super-resolution microscopy (STORM), we define the gap between intersecting MTs and the liposome azimuthal approach angle heading into the intersection. We develop an in silico model in which kinesin-1 motors diffuse on the liposome surface, simultaneously engage the intersecting MTs, generate forces and detach from MTs governed by the motors' mechanochemical cycle, and undergo a tug-of-war with the winning team determining the directional outcome in 3D. The model predicts that 1-3 motors typically engage the MT, consistent with optical trapping measurements. Modeled liposomes also predominantly go straight through 3D intersections over a range of intersection gaps and liposome approach angles, even when obstructed by the crossing MT. Our observations and modeling offer mechanistic insights into how cells might tune the MT cytoskeleton, cargo, and motors to modulate cargo transport.