We show that d+1-dimensional surface growth models can be mapped onto driven lattice gases of d-mers. The continuous surface growth corresponds to one dimensional drift of d-mers perpendicular to the (d-1-dimensional "plane" spanned by the d-mers. This facilitates efficient bit-coded algorithms with generalized Kawasaki dynamics of spins. Our simulations in d=2, 3, 4, 5 dimensions provide scaling exponent estimates on much larger system sizes and simulations times published so far, where the effective growth exponent exhibits an increase. We provide evidence for the agreement with field theoretical predictions of the Kardar-Parisi-Zhang universality class and numerical results. We show that the (2+1)-dimensional exponents conciliate with the values suggested by Lässig within error margin, for the largest system sizes studied here, but we cannot support his predictions for (3+1)d numerically.