A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. We show that for an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. Using this equivalence, we show that the simple random walk on any distributional lattice in a nonamenable space has positive embedded speed. For nonpositively curved, simply connected spaces, we show that the simple random walk on a Poisson--Voronoi tessellation has positive graph speed by developing some additional structure for Poisson--Voronoi tessellations.