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An even-cycle decomposition of a graph G is a partition of E(G) into cyclesof even length. Evidently, every Eulerian bipartite graph has an even-cycledecomposition. Seymour (1981) proved that every 2-connected loopless Eulerianplanar graph with an even number of edges also admits an even-cycledecomposition. Later, Zhang (1994) generalized this to graphs with no$K_5$-minor.  Our main theorem gives sufficient conditions for the existence of even-cycledecompositions of graphs in the absence of odd minors. Namely, we prove thatevery 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an evennumber of edges has an even-cycle decomposition.  This is best possible in the sense that `odd-$K_4$-minor-free' cannot bereplaced with `odd-$K_5$-minor-free.' The main technical ingredient is astructural characterization of the class of odd-$K_4$-minor-free graphs, whichis due to Lov\'asz, Seymour, Schrijver, and Truemper.

Even-cycle decompositions of graphs with no odd-$K_4$-minor