This paper proposes a new Distributed Moving Horizon Estimation (DMHE) algorithm for state estimation of a discrete-time linear system by a sensor network. The main contribution consists in using a pre-estimating Luenberger observer in the formulation of the local problem to be solved by each sensor. This results in a significant reduction of the computation time while preserving at the same time the accuracy of the estimates. The state estimate computed by each sensor is capable to converge even under weak observability conditions. Comparison with existing approaches is proposed through a simulation example, for both centralized and distributed cases, in terms of estimation accuracy and computation time.