We use the truncated Wigner approximation (TWA) to study the quench dynamics of a two-dimensional lattice system composed of spinless fermions interacting with an underlying disorder. First, we show that semiclassical dynamics typically relax faster than full quantum dynamics. This result is obtained by comparing semiclassical mechanics with exact diagonalization and Lanczos propagation of one-dimensional chains. We then exploit the TWA capability to simulate large lattices to investigate how the relaxation rate depends on the dimensionality of the system under study. We show that strongly disordered one- and two-dimensional systems exhibit transient logarithmic time relaxation. This was recently established in one-dimensional chains. Such relaxation corresponds to the notorious $1/f$ noise in strong disorder.