We investigate the energy-momentum deposition due to neutrino-antineutrino annihilation in the vicinity of axisymmetric, accreting black holes (BHs) by numerically ray-tracing neutrino trajectories in a Kerr space-time. Hyperaccreting stellar-mass BHs are widely considered as energy sources that can drive ultrarelativistic outflows with the potential to produce gamma-ray bursts. In contrast to earlier works, we provide an extensive and detailed parameter study of the influence of general relativistic (GR) effects and of different neutrinosphere geometries. These include idealized thin disks, tori, and spheres, or are constructed as non-selfgravitating equilibrium matter distributions for varied BH rotation. Considering isothermal neutrinospheres with the same temperature and surface area, we confirm previous results that compared to Newtonian calculations, GR effects increase the annihilation rate measured by an observer at infinity by a factor of 2 when the neutrinosphere is a disk. However, in case of a torus and a sphere the influence of GR effects is globally only ~25%, although locally it can be significantly larger. Independent of whether GR effects are included, disks yield the highest energy deposition rates, followed by tori and, with the lowest rates, spheres. For disks and tori, increasing the angular momentum of the BH from 0 to 1 enhances the energy deposition rate measured by an observer at infinity by roughly a factor of 2 due to the shrinking inner radius of the neutrinosphere. (abridged)