This paper shows that the multicommodity flow problem on a class of planar undirected graphs can be reduced to another famous combinatorial problem, the weighted matching problem. Assume that in a given planar graph G all the sources can be joined to the corresponding sinks without destroying the planarity. Then we show that the feasibility of multicommodity flows can be tested simply by solving, once, the weighted matching problem on a certain graph constructed from G, and that the multicommodity flows of given demands can be found by solving the matching problem $O(n)$ times if G has n vertices. Efficient algorithms are also given for detecting negative and minimum cycles in planar undirected graphs.