Belief Propagation (BP) allows to approximate exact probabilistic inference in graphical models, such as Markov networks (also called Markov random fields, or undirected graphical models). However, no exact convergence guarantees for BP are known, in general. Recent work has proposed to approximate BP by linearizing the update equations around default values for the special case when all edges in the Markov network carry the same symmetric, doubly stochastic potential. This linearization has led to exact convergence guarantees, considerable speed-up, while maintaining high quality results in network-based classification (i.e. when we only care about the most likely label or class for each node and not the exact probabilities). The present paper generalizes our prior work on Linearized Belief Propagation (LinBP) with an approach that approximates Loopy Belief Propagation on any pairwise Markov network with the problem of solving a linear equation system.