We study numerically the behavior of the Biham-Middleton-Levine traffic model in three dimensions. Our extensive numerical simulations show that the phase diagram for this model in three dimensions is markedly different from that in one and two dimensions. In addition to the full speed moving as well as the completely jamming phases, whose respective average asymptotic car speeds (v) equal one and zero, we observe an extensive region of car densities rho with a low but nonzero average asymptotic car speed. The transition from this extensive low average asymptotic car speed region to the completely jamming region is at least second order. We argue that this low speed region is a result of the formation of a spatially limited-extended percolating cluster. Thus, this low speed phase is present in the (n>3)-dimensional Biham-Middleton-Levine model as well.