In this thesis, we establish a new link between Poisson Geometry and Combinatorics. We introduce the notion of tropicalization for Poisson structures on R^n with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to C^n viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus. The main example considered in the thesis is the canonical Poisson bracket on the dual Poisson-Lie group G^* for G=U(n) in the cluster coordinates of Fomin-Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand-Zeitlin completely integrable system of Guillemin-Sternberg and Flaschka-Ratiu.