The control of the surface wettability is of great interest for technological applications as well as for the fundamental understanding of surface phenomena. In this article, we describe the dissolution behavior of droplets wetting a micropatterned surface consisting of smooth concentric circular grooves. In the experiments, a droplet of alcohol (1-pentanol) is placed onto water-immersed micropatterns. When the drops dissolve, the dynamics of the receding contact line occurs in two different modes. In addition to the stick-jump mode with jumps from one ring to the next inner one, our study reveals a second dissolution mode, which we refer to as zipping-depinning. The velocity of the zipping-depinning fronts is governed by the dissolution rate. At the early stage of the droplet dissolution, our experimental results are in good agreement with the theoretical predictions by Debuisson et al. [ Appl. Phys. Lett. 2011, 99, 184101]. With an extended model, we can accurately describe the dissolution dynamics in both stick-jump and zipping-depinning modes.