This work considers a special class of consensus seeking networks that we term Eulerian consensus networks. These consensus networks are defined over the class of graphs known as Eulerian, and admit many combinatorial features that are beneficial for both analysis and synthesis purposes. We first consider the H2 performance of Eulerian consensus systems and reveal that the performance is related to the length of cycles in the graph. The structure of Eulerian graphs motivates a procedure for designing large-scale Eulerian consensus networks. We propose a suite of algorithms for constructing these networks and demonstrate their numerical effectiveness on a graph with 5000 nodes.