We present a new systematic set of algorithms for interpolated variational transition-state theory by mapping (IVTST-M). In this method, which is designed to allow efficient direct dynamics calculations, rate constants for chemical reactions are evaluated by variational transition-state theory with multidimensional tunneling approximations based on reaction-path data. The data (energies, energy gradients, and Hessians) are computed at a small number of points along a reaction path and fitted to splines under tension as functions of a mapped independent variable that is a nonlinear function of the reaction coordinate. The theory is illustrated and tested by several examples, and standard choices are employed for all parameters and functional forms to provide a realistic test of how the method might perform when applied as an automatic scheme without fine-tuning each reaction. For eight test cases, we obtain reasonable accuracy (as compared to calculations with the same potential surface with the reaction path followed as far as necessary for convergence) with Hessians at only six nonstationary points.