We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato-Rosenblum. It also generalizes several results concerning perturbations of the discrete (free or periodic) Schrödinger operator. The paper concerns "one-sided" Jacobi operators (i.e. in l²(ℕ)) and is based on the method of subordinacy. We provide some spectral results for the unperturbed, periodic case, and also an appendix containing some subordination theory tools.