We propose a new iterative distributed algorithm for linear minimum mean-squared-error (LMMSE) estimation in sensor networks whose measurements follow a Gaussian hidden Markov graphical model with cycles. The embedded polygons algorithm decomposes a loopy graphical model into a number of linked embedded polygons and then applies a parallel block Gauss-Seidel iteration comprising local LMMSE estimation on each polygon (involving inversion of a small matrix) followed by an information exchange between neighboring nodes and polygons. The algorithm is robust to temporary communication faults such as link failures and sleeping nodes and enjoys guaranteed convergence under mild conditions. A simulation study indicates that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep. Thus, somewhat surprisingly, energy conservation strategies such as low-powered transmission and aggressive sleep schedules could actually be counterproductive.