Discrete ordinates radiation transport problems in two-dimensional geometry that have curvilinear (or inclined) surfaces are not accurately represented with rectangular cells. Some such problems do not have the symmetry necessary for cylindrical (r,) geometry. We have developed various spatial quadratures for arbitrarily-shaped triangular cells, and an algorithm for tracking the flow from cell to cell through an unstructured grid of such cells. Such a grid can accurately approximate curved or inclined surfaces. Furthermore, unstructured grids can increase efficiency by providing mesh refinement only where needed. By splitting each cell along the streaming direction, into two triangular subcells, each of which has only one inflow edge and one outflow edge, a wide variety of spatial quadratures can be readily adapted to these grids. We present split-cell versions of the step, step characteristic, diamond difference, and linear characteristic spatial quadratures and necessary details of implementation of the flow-tracking algorithm. Test results, also presented here, demonstrate advantages of unstructured triangular grids and show the robustness of the method for grids formed by Delaunay triangulation of randomly-chosen vertices. Indeed, randomness of the mesh is observed to ameliorate the systematic accumulation of spatial truncation errors, thus decreasing numerical diffusion.