The Krogh cylinder model for oxygen diffusion from capillaries is the starting point for many theoretical models of oxygen delivery to tissue. It assumes an idealized geometry, in which the capillaries are identical, parallel, and evenly spaced. We present theoretical simulations of oxygen delivery to tissue by networks of microvessels, under less restrictive geometrical assumptions. These simulations are carried out using a Green's function approach. Three vascular geometries are considered: a single capillary passing near an arteriole; a configuration of arterioles and capillaries based on observations of skeletal muscle; and a network of microvessels based on observations of a tumor preparation. In these simulations, several phenomena emerge that are not seen in the Krogh model. These phenomena fall into two main classes: diffusive interactions between neighboring vessel segments; and effects of heterogeneous vessel spacing. Examples of these phenomena are described, and their significance for the function of normal tissues and for the treatment of tumors is briefly discussed.