  We define Poisson-geometric analogues of Kitaev's lattice models. They are
obtained from a Kitaev model on an embedded graph $\Gamma$ by replacing its
Hopf algebraic data with Poisson data for a Poisson-Lie group G.
  Each edge is assigned a copy of the Heisenberg double $\mathcal H(G)$. Each
vertex (face) of $\Gamma$ defines a Poisson action of $G$ (of $G^*$) on the
product of these Heisenberg doubles. The actions for a vertex and adjacent face
form a Poisson action of the double Poisson-Lie group $D(G)$. We define Poisson
counterparts of vertex and face operators and relate them via the Poisson
bracket to the vector fields generating the actions of $D(G)$.
  We construct an isomorphism of Poisson $D(G)$-spaces between this
Poisson-geometrical Kitaev model and Fock and Rosly's Poisson structure for the
graph $\Gamma$ and the Poisson-Lie group $D(G)$. This decouples the latter and
represents it as a product of Heisenberg doubles. It also relates the
Poisson-geometrical Kitaev model to the symplectic structure on the moduli
space of flat $D(G)$-bundles on an oriented surface with boundary constructed
from $\Gamma$.
