We solve the Holstein equation describing radiation trapping in an atomic vapor by the geometric quantization technique. The treatment is based on studying the integral trapping equation as a wave equation for an associated quasiparticle with a complicated form of its dispersion law. The latter is determined by the spectral properties of the vapor medium. Extending our previous work, which dealt with one-dimensional (1D) geometries, we consider here more realistic two- and three-dimensional (2D and 3D) vapor cell geometries. For this, an explicit representation of the phase factors is derived when the quasiparticle is reflected under arbitrary angles from the surface of the cell confining the vapor. We give closed-form equations to obtain analytically the complete 3D spectrum of escape factors in finite cylinder, sphere, parallelepiped, and finite prismlike geometries for all practically occurring line shapes and opacities. In addition, the relevant semiclassical variational method is applied to develop a perturbation theory for the evaluation of escape factors in more complicated geometries that are sometimes adopted in experiments, including infinite (or finite) elliptical cylinders, prolate and oblate ellipsoids. Comparisons with numerical calculations show that results are accurate within [Formula Presented] for the ground modes and even better for higher-order modes.