We study the effect of the connectivity pattern of complex networks in the propagation dynamics of epidemics. The growth time scale of outbreaks is inversely proportional to the network degree fluctuations, signaling that epidemics spread almost instantaneously in networks with scale-free degree distributions. This feature is associated to an epidemic propagation that follows a precise hierarchical dynamics. Once reached the highly connected hubs, the infection pervades the network in a progressive cascade across smaller degree classes. The present results are relevant for the development of adaptive containment strategies.