Multiple description (MD) coding is a source coding technique for information transmission over unreliable networks. In MD coding, the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions. In this paper, we study the problem of MD coding of stationary Gaussian sources with memory. First, we compute an approximate MD rate distortion region for these sources, which we prove to be asymptotically tight at high rates. This region generalizes the MD rate distortion region of El Gamal and Cover (1982), and Ozarow (1980) for memoryless Gaussian sources. Then, we develop an algorithm for the design of optimal two-channel biorthogonal filter banks for MD coding of Gaussian sources. We show that optimal filters are obtained by allocating the redundancy over frequency with a reverse "water-filling" strategy. Finally, we present experimental results which show the effectiveness of our filter banks in the low complexity, low rate regime