We first introduce the conjugate linearized Ricci flow, a version of the linearization of the Ricci flow motivated by Perelman’s treatment of Ricci flow as a gradient flow. The flow takes into account the diffeomorphism invariance of the Ricci flow, which is the only obstruction to its strict parabolicity. We characterize the conjugate linearized Ricci flow on closed three-manifolds of bounded geometry and discuss its properties. In particular, we express the evolution of the metric and of its Ricci tensor in terms of the backward heat kernel of the conjugate linearized Ricci flow. These results provide various conservation laws and monotonicity formulas for the linearized flow. These results may be of interest to both analytical treatment of Ricci flow and physical applications of Ricci flow.