In this paper, we introduce a new heuristic algorithm for finding maximum planar subgraphs of a graph. Our algorithm is based on the simulated annealing optimization scheme. We compare the quality of the solutions and running times of our algorithm with greedy randomized adaptive search procedure (GRASP) and branch-and-cut heuristics. We show that simulated annealing is an efficient method to find planar subgraphs with a large number of edges. The algorithm clearly outperforms earlier heuristic algorithms; it is faster and its solution quality is better. The algorithm is very simple, although it needs an implementation for the planarity test.