The motion of small cargo particles on microtubules by means of motor proteins in disordered microtubule networks is investigated theoretically using both analytical tools and computer simulations. Different network topologies in two and three dimensions are considered, one of which has been recently studied experimentally by Salman [Biophys. J. 89, 2134 (2005)]. A generalization of the random velocity model is used to derive the mean-square displacement of the cargo particle. We find that all cases belong to the class of anomalous superdiffusion, which is sensitive mainly to the dimensionality of the network and only marginally to its topology. Yet in three dimensions the motion is very close to simple diffusion, with sublogarithmic corrections that depend on the network topology. When details of the thermal diffusion in the bulk solution are included, no significant change to the asymptotic time behavior is found. However, a small asymmetry in the mean microtubule polarity affects the corresponding long-time behavior. We also study a three-dimensional model of the microtubule network in living animal cells. Three first-passage-time problems of intracellular transport are simulated and analyzed for different motor processivities: (i) cargo that originates near the nucleus and has to reach the membrane, (ii) cargo that originates from the membrane and has to reach the nucleus, and (iii) cargo that leaves the nucleus and has to reach a specific target in the cytoplasm. We conclude that while a higher motor processivity increases the transport efficiency in cases (i) and (ii), in case (iii) it has the opposite effect. We conjecture that the balance between the different network tasks, as manifested in cases (i) and (ii) versus case (iii), may be the reason for the evolutionary choice of a finite motor processivity.