The paper presents a new semiclassical theory of multidimensional tunneling and its application to the decay problem. A simple semiclassical expression for the decay rate constant is derived in terms of complex valued family of classical trajectories in the decay valley while the algebraic form of the family in the tunneling region is found by means of classical canonical perturbation theory. This provides a tool to analytically continue the classical trajectories into the decay valley, with a numerical illustration of such a continuation being done for a model 2D decay rate problem. The calculated results for the rate constant are found to be in good agreement with the exact ones for the high levels where the suggested perturbative treatment is applicable. It is also shown that the formulated theory can be directly compared with a previously proposed hopping method which gives a way to examine the accuracy of the latter without exact quantum calculations.