A method that computes minimum energy paths (MEPs) of wetting transitions is developed. The method couples the Cahn-Hilliard formulation of a modified phase-field method with the simplified string method. Its main computational kernel is the fast Fourier transform that is efficiently performed on graphics processing units. The effectiveness of the proposed method is demonstrated on two types of transitions of droplets on grooved surfaces. The first is the transition from the Cassie-Baxter wetting state to the Wenzel state, where it is shown that it progresses in a sequential manner with the droplet wetting each groove successively. The second transition type is a lateral displacement of the droplet against the grooves, where the droplet successively detaches/attaches from/to the rear/front protrusion of the surface (a transition in the reverse order is also possible). The energy barriers of both the transitions are extracted from the MEP; they are useful for the evaluation of the robustness of superhydrophobic surfaces (resistance to the Cassie-Baxter to Wenzel transition) and the droplet mobility on those surfaces (high mobility/small resistance to lateral displacements). The relation of the MEP with the potential transition paths coming from the solution space mapping is discussed.