On a lossless periodic dielectric structure sandwiched between two homogeneous media, bound states in the continuum (BICs) with real frequencies and real Bloch wave vectors may exist, and they decay exponentially in the surrounding homogeneous media and do not couple with propagating plane waves with the same frequencies and wave vectors. The BICs are of significant current interest because they give rise to high-Q resonances when the structure or the Bloch wave vector is slightly perturbed. In this paper, the effect of a small material loss on the BICs is analyzed by a perturbation method and illustrated by numerical results. It is shown that bound states with complex frequencies near the continuum appear, but they behave differently depending on whether the BIC is symmetry protected or not. The Bloch wave vector of a bound state with a complex frequency can be real if the original BIC is symmetry protected, and it is usually complex if the original BIC is not symmetry protected. Our study improves the theoretical understanding of BICs and provides useful insight for their practical applications.