A recently introduced real-space renormalization-group technique; developed for the analysis of processes in the Kardar-Parisi-Zhang universality class; is generalized and tested by applying it to a different family of surface-growth processes. In particular; we consider a growth model exhibiting a rich phenomenology even in one dimension. It has four different phases and a directed percolation-related roughening transition. The renormalization method reproduces extremely well all of the phase diagram; the roughness exponents in all the phases; and the separatrix among them. This proves the versatility of the method and elucidates interesting physical mechanisms.