We consider energy harvesting communications over an intersymbol interference (ISI) channel corrupted by additive white Gaussian noise. We assume that the energy arrivals at the transmitter, which is equipped with a finite battery, are independent and identically distributed, and provide a method of computing achievable information rates with channel inputs selected from a finite alphabet. We illustrate the methodology via a set of examples quantifying the effects of energy arrival rates as well as battery capacity on the achievable information rates.