We deal with distributed linear minimum mean-square error (LMMSE) estimation of a random signal vector based on observations collected across a wireless sensor network (WSN). We cast this decentralized estimation problem as the solution of multiple constrained convex optimization subproblems. Using the method of multipliers in conjunction with a block coordinated descent approach we demonstrate how the resultant algorithm can be decomposed into a set of simpler tasks suitable for distributed implementation. Relative to existing alternatives, we establish that the novel decentralized algorithm guarantees convergence to the centralized LMMSE estimator for quite general (possibly nonlinear and/or non-Gaussian) data models. Through numerical experiments, we finally illustrate the convergence properties of the algorithm.