A systematic understanding of the miscibility of grafted nanorods in polymer melt is required in order to synthesize materials and make devices with controllable properties. While there have been a number of studies on the effect of graft length and graft density on the miscibility of grafted nanorods, the effect of graft arrangement and nanorod geometry remains to be explored. We use integral equation theory to study the dispersion, macrophase separation, and self-assembly of sparsely grafted nanorods in polymer melt. This phase behavior is studied as a-function of nanorod diameter, aspect ratio, and density as well as the length and arrangement of the polymer grafts. The phase behavior of these systems is a result of a competition between matrix-induced depletion attraction between nanorods and the steric stabilization provided by grafts. Because of steric shielding of the grafts, nanorod miscibility usually increases with graft length, and trans-grafted rods are more soluble than cis-grafted rods. Depletion attraction is stronger between larger nanorods. Therefore, shorter and thinner nanorods with longer grafts are found to microphase separate, while longer and thicker nanorods with shorter grafts tend to macrophase separate from the matrix polymer. While miscibility of bare nanorods is a monotonically decreasing function of aspect ratio, the miscibility of grafted nanorods can also be a nonmonotonic or a monotonically increasing function of aspect ratio. Grafted nanorods become less soluble in the matrix polymer as their diameter increases. Thus, the effect of rod geometry on the phase behavior of these composites is subtle and complex.