The time-constrained packet routing problem is to schedule a set of packets to be transmitted through a multi-node network, where every packet has a source and a destination (as in traditional packet routing problems) as well as a release time and deadline. The objective is to schedule the maximum number of packets subject to deadline constraints. This problem is studied in [1], where it is shown that the problem is NP-Complete even when the underlying topology is a linear array. Approximation algorithms are also provided in [1] for the linear array and the unidirectional ring for both the case where packets may be buffered in transit and the case where they may not be. In this paper we extend the results of [1] in two directions. First, we consider the more general network topologies of trees and 2-dimensional meshes. Second, we associate with each packet a measure of utility, called a weight, and study the problem of maximizing the total weight of the packets that are scheduled subject to their timing constraints. For the bufferless case, we provide constant factor approximation algorithms for the time-constrained scheduling problem with weighted packets on trees and mashes. We also provide logarithmic approximations for the same problems in the buffered case. These results are complemented by new lower bounds, which demonstrate that we cannot hope to achieve the same results for general network topologies. For example, we show that if k packets are require to follow prescribed paths in an arbitrary graph, then unless NP = ZPP, there is no polynomial-time k1-ε-approximation, for any ε > 0, to the optimal set of packets that can be scheduled.