A solid-on-solid model for {111} surfaces of diamond-cubic materials that correctly takes into account the diamond-cubic crystal structure has been developed for Monte Carlo simulation. In addition to a roughening transition at temperature TR, a distinct preroughening transition at TPR≈0.43TR is indicated by divergences in the surface specific heat and order-parameter susceptibility. Preroughening appears to arise naturally in our nearest-neighbor bond model from the entropic freedom available in the nontrivial crystal structure. Preroughening is shown to dramatically lower the nucleation barrier for growth and etching at low driving forces.