A Cartesian grid method with solution-adaptive anisotropic refinement and coarsening is developed for simulating time-dependent incompressible flows. The Cartesian grid cells and faces are managed using an unstructured data approach, and algorithms are described for the time-accurate transient anisotropic refinement and coarsening of the cells. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure, and advanced in time using the fractional step method. Significant savings in the memory requirement of the method can be realized by advancing the velocity field using a novel approximate factorization technique, although an iterative technique is also presented. The pressure Poisson equation is solved using additive correction multigrid, and an efficient coarse grid selection algorithm is presented. Finally, the Cartesian cell geometry allows the development of relatively simple analytic expressions for the optimal cell dimensions based on limiting the velocity interpolation error. The overall method is validated by solving several benchmark flows, including the 2D and 3D lid-driven cavity flows, and the 2D flow around a circular cylinder. In this latter case, an immersed boundary method is used to handle the embedded cylinder boundary.