Applying an electric field to polarizable colloidal particles, whose permittivity differs from that of the dispersing medium, generates induced dipoles that promote the formation of string-like clusters and ultimately alter the fluid mechanical and rheological properties. Complex systems of this kind, whose electric-field-induced rheology can be manipulated between that of viscous and elastic materials, are referred to as electrorheological fluids. By using dynamic Monte Carlo simulations, we investigate the dynamics of self-assembly of dielectric nanocubes upon application of an electric field. Switching the field on induces in-particle dipoles and, at sufficiently large field intensity, leads to string-like clusters of variable length across a spectrum of volume fractions. The kinetics of switching from the isotropic to the string-like state suggests the existence of two mechanisms, the first related to the nucleation of chains and the second to the competition between further merging and separation. We characterize the transient unsteady state by following the chain length distribution and analyzing the probability of the transition of nanocubes from one chain to another over time. Additionally, we employ passive microrheology to gain insight into the effect of the electric field on the viscoelastic response of our model fluid. Not only do we observe that it becomes more viscoelastic in the presence of the field but also that its viscoelasticity assumes an anisotropic signature, with both viscous and elastic moduli in planes perpendicular to the external field being larger than those along it.