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Adaptive Mesh Refinement (AMR) is a technique used in numerical simulations to automatically refine (or de-refine) certain regions of the physical domain in a finite difference calculation. AMR data consists of nested hierarchies of data grids. As AMR visualization is still a relatively unexplored topic, our work is motivated by the need to perform efficient visualization of large AMR data sets. We present a software algorithm for parallel direct volume rendering of AMR data using a cell-projection technique on several different parallel platforms. Our algorithm can use one of several different distribution methods, and we present performance results for each of these alternative approaches. By partitioning an AMR data set into blocks of constant resolution and estimating rendering costs of individual blocks using an application specific benchmark, it is possible to achieve even load balancing.