  This paper investigates solutions of hyperbolic diffusion equations in
$\mathbb{R}^3$ with random initial conditions. The solutions are given as
spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are
studied. All assumptions are formulated in terms of the angular power spectrum
or the spectral measure of the random initial conditions. Approximations to the
exact solutions are given. Upper bounds for the mean-square convergence rates
of the approximation fields are obtained. The smoothness properties of the
exact solution and its approximation are also investigated. It is demonstrated
that the H\"{o}lder-type continuity of the solution depends on the decay of the
angular power spectrum. Conditions on the spectral measure of initial
conditions that guarantee short or long-range dependence of the solutions are
given. Numerical studies are presented to verify the theoretical findings.
