We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by (t + 1) + cy(t) = f (y(t)), where f: ℝ → ℝ and β > 0 is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant uf (u) > 0 such that |u| ≥ β whenever c = 1. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: |b| < 2, N across-1(-b/2), and π is an even multiple of c ≠ 0.