This paper proposes a framework for joint source-channel decoding of Markov sequences that are encoded by an entropy coded multiple description quantizer (MDQ), and transmitted via a lossy network. This framework is particularly suited for lossy networks of inexpensive energy-deprived mobile source encoders. Our approach is one of maximum a posteriori probability (MAP) sequence estimation that exploits both the source memory and the correlation between different MDQ descriptions. The MAP problem is modeled and solved as one of the longest path in a weighted directed acyclic graph. For MDQ-compressed Markov sequences impaired by both bit errors and erasure errors, the proposed joint source-channel MAP decoder can achieve 5dB higher SNR than the conventional hard-decision decoder. Furthermore, the new MDQ decoding technique unifies the treatments of different subsets of the K descriptions available at the decoder, circumventing the thorny issue of requiring up to 2K - 1 MDQ side decoders.