Undirected graphical models, such as Gaussian graphical models, Ising, and multinomial/categorical graphical models, are widely used in a variety of applications for modeling distributions over a large number of variables. These standard instances, however, are ill-suited to modeling count data, which are increasingly ubiquitous in big-data settings such as genomic sequencing data, user-ratings data, spatial incidence data, climate studies, and site visits. Existing classes of Poisson graphical models, which arise as the joint distributions that correspond to Poisson distributed node-conditional distributions, have a major drawback: they can only model negative conditional dependencies for reasons of normalizability given its infinite domain. In this paper, our objective is to modify the Poisson graphical model distribution so that it can capture a rich dependence structure between count-valued variables. We begin by discussing two strategies for truncating the Poisson distribution and show that only one of these leads to a valid joint distribution; even this model, however, has limitations on the types of variables and dependencies that may be modeled. To address this, we propose two novel variants of the Poisson distribution and their corresponding joint graphical model distributions. These models provide a class of Poisson graphical models that can capture both positive and negative conditional dependencies between count-valued variables. One can learn the graph structure of our model via penalized neighborhood selection, and we demonstrate the performance of our methods by learning simulated networks as well as a network from microRNA-Sequencing data.