Using concepts from transient chaos and stochastic dynamics, we develop a perturbative solution for multidimensional activated rate processes. The solution is applicable to the underdamped regime where system dynamics prevails over bath fluctuations. The baseline of the method is the partition of the multidimensional reactive flux in a chaotic system to a sum of independent fluxes in one-dimensional systems. The partition is based on the underlying dynamics of the multidimensional system. The method is fast and explains the high and low temperature dependence of multidimensional reaction rates.