Recently, new opportunities for utilizing extremely high frequencies have become instrumental to designing the fifth-generation mobile technology. The use of highly directional antennas in millimeter-wave (mm-wave) bands poses an important question of whether 2D modeling suffices to capture the resulting system performance accurately. In this paper, we develop a novel mathematical framework for performance assessment of the emerging 3D mm-wave communication scenarios, which takes into account vertical and planar directivities at both ends of a radio link, blockage effects in three dimensions, and random heights of the communicating entities. We also formulate models having different levels of details and verify their accuracy for a wide range of system parameters. We show that capturing the randomness of both the transmitting and receiving heights as well as the vertical antenna directivities becomes crucial for accurate system characterization. The conventional planar models provide overly optimistic results that overestimate performance. For instance, the model with fixed heights that disregards the effect of vertical exposure is utterly pessimistic. The other two models, one having random heights and neglecting vertical exposure and another one characterized by fixed heights and capturing vertical exposure are less computationally expensive and can be used as feasible approximations for certain ranges of input parameters.