We study the problem of scheduling n independent general multiprocessor tasks on a fixed number of processors, where the objective is to compute a non-preemptive schedule minimizing the average weighted completion time. For each task, its execution time is given as a function of the subset of processors assigned to the task. We propose here a polynomial-time approximation scheme for the problem that computes a (1 + ε)-approximate solution in O(n log n) time for any fixed ε > 0 accuracy. This provides a generalization and integration of some recent polynomial-time approximation schemes for scheduling jobs on unrelated machines [1,18] and multiprocessor tasks on dedicated processors [2], respectively, with the average weighted completion time objective, since the latter models are included as special cases in our problem.