We present an efficient raycasting-based rendering algorithm for view-dependent piecewise constant representations of volumetric data. Our algorithm leverages the properties of perspective projection to simplify intersections of rays with the view-dependent frustums that form part of these representations. It also leverages spatial homogeneity in the underlying volume data to minimize memory accesses. We further introduce techniques for skipping empty-space and for dynamic subsampling for accelerated approximate renderings at controlled frame rates. Benchmarks show that responsive frame rates can be achieved close to the viewpoint of generation for HD display resolutions, while providing high-fidelity approximate renderings of Gigabyte-sized volumes.