We study the problem of multiple description (MD) coding of stationary Gaussian sources with memory. First, we compute an approximate rate distortion region for these sources, which we prove to be asymptotically tight at high rates: this region generalizes the standard MD rate distortion region for memoryless sources. Then we develop an algorithm for the design of optimal biorthogonal filter banks for MD coding. Finally. We present some experimental results, where we measure the deviation from optimality of our proposed system. For almost uncorrelated sources the gap between the performance of our proposed system and the ideal bounds is quite high, on the other hand for highly correlated sources this gap is reduced, due to the ability of our system to take advantage of the memory in the source. In this case, in realistic scenarios where finite complexity/delay is an issue, the subband coding approach is competitive with other approaches such as the decorrelating transform followed by MD scalar quantizers.