This paper is concerned with the linear minimum mean square error estimation (LMMSE) for the multi-rate sampling systems with multi-channel observation delays and unknown Markovian packet losses. The original system is firstly transformed into a single-rate jumping parameter system with multi-channel and delay-free observations by employing the lifting technique and introducing a set of reorganized observations and Markov chains. Then, the single-rate system is converted into a general linear system without delays by defining a new group of extended states. Based on the innovation analysis method, a liner minimum mean square error estimator is developed, and the estimator gain is obtained in terms of generalized Riccati difference equations based on a set of coupled Lyapunov equations. Therefore, the original state estimation problem is solved via the jumping parameter property. Finally, the convergence of the Riccati equation is analyzed and a stationary filter is obtained. The novelty of this paper lies in the introduction of the reorganized observations and multi-state Markov chains.