A path-integral (PI) approach to real-time quantum dynamics is developed which is suitable to treat the short-time dynamics of vibronic-coupling systems involving many degrees of freedom. The theory is formulated for the case of two electronic states which are coupled by a single active vibrational mode and whose energy separation is modulated by many so-called tuning modes. Time-dependent correlation functions are expressed as sums over ah possible paths in the space of two electronic states in discretized time. For each electronic path, the multi-mode vibrational propagator factorizes into a product of single-mode propagators. Introducing the concept of classes of approximately equivalent paths, the summation over paths is replaced by a summation over classes and the computation of propagator averages within each class. It is shown that the propagator averages can efficiently be calculated by a recursive scheme. The performance of the PI method has been tested for a two-state four-mode model representing S-1-S-2 vibronic coupling in pyrazine. The PI results (time-dependent correlation functions and absorption spectral are compared with numerically exact reference data which are available for this model. To demonstrate the potential of the path-integral approach for multi-mode problems, calculations are reported for a twenty-four-mode vibronic-coupling model.