A new instanton approach is reported to tunneling at zero-temperature in multidimensional (MD) systems in which a "light particle" is transferred between two equivalent "heavy" sites. The method is based on two concepts. The first is that an adequate MD potential energy surface can be generated from input of the stationary configurations only, by choosing as a basis the normal modes of the transition state. It takes the form of a double-minimum potential along the mode with imaginary frequency and coupling terms to the remaining (harmonic) oscillators. Standard integrating out of the oscillators gives rise to an effective 1D instanton problem for the adiabatic potential, but requires evaluation of a nonlocal term in the Euclidean action, governed by exponential (memory) kernels. The second concept is that this nonlocal action can be treated as a "perturbation," for which a new approximate instanton solution is derived, termed the "rainbow" solution. Key to the approach is avoidance of approximations to the exponential kernels, which is made possible by a remarkable conversion property of the rainbow solution. This leads to a new approximation scheme for direct evaluation of the Euclidean action, which avoids the time-consuming search of the exact instanton trajectory. This "rainbow approximation" can handle coupling to modes that cover a wide range of frequencies and bridge the gap between the adiabatic and sudden approximations. It suffers far fewer restrictions than these conventional approximations and is proving particularly effective for systems with strong coupling, such as proton transfer in hydrogen bonds. Comparison with the known exact instanton action in two-dimensional models and application to zero-level tunneling splittings in two isotopomers of malonaldehyde are presented to show the accuracy and efficiency of the approach.