This paper investigates three categories of algorithms for direct volume rendering of unstructured grids, which are image-space, object-space, and hybrid methods. We propose three new algorithms. Cell Projection (CP) algorithm, which falls into object-space category, is capable of rendering non-convex meshes through a simple yet efficient sorting schema that exploits both image and object space coherencies. Existing hybrid methods use object-then-image traversal order that enforces the processing of each cell. Thus, these algorithms perform redundant operations and do not support early ray termination. We propose a hybrid method, called Span-Buffer Ray Casting (SBRC), that can support early ray termination discarding redundant operations by employing image-then-object traversal order. Another hybrid method, called Koyamada-SBRC (K-SBRC), is proposed with the motivation of refining image-space and hybrid methods to extract the best features of them. This method is developed by blending SBRC approach with Koyamada's algorithm, which is an efficient image-space algorithm. All proposed algorithms are capable of handling acyclic non-convex meshes and generating images of acceptable quality. SBRC and K-SBRC algorithms have the additional capabilities of rendering cyclic meshes and supporting early ray termination. The proposed algorithms and Koyamada's algorithm are implemented and experimented in a common framework for analyzing their relative performance.