A modular system architecture based on separate compression and network coding is known to be theoretically suboptimal for relevant classes of sensor networks with correlated sources. Motivated by this observation, we present a feasible solution for joint source and network coding with distortion constraints. By choosing encoders that are simple scalar index assignments, we are able to move the complexity to the destination decoder. Given the network topology and the correlation structure of the data, our algorithms solve the problem of finding encoder and decoder instances that minimize the mean square error of every sample. A proof-of-concept and the complexity analysis of the proposed algorithms underline the effectiveness of our factor graph approach. The presented schemes are shown to yield low-distortion estimates of the collected data even in scenarios where a modular solution would fail.