The paper addresses the problem of designing distributed observers for discrete linear time-invariant (LTI) systems with distributed sensor nodes subjected to bounded measurement noise. A solution is proposed in terms of a distributed LTI Luenberger observer, thus departing from common linear time-varying solutions rooted in consensus-based distributed estimation techniques, and dispensing with the need for the exchange of covariance matrices. It is shown, under the conditions of collective observability, strong connectivity of the sensor communication network, and invertibility of the state transition matrix, that the resulting observer yields ultimate boundedness of the estimation error. A design example is given where the asymptotic performance of the proposed observer is shown to be similar to that obtained using a time-varying distributed Kalman filtering approach.