Random Threshold Networks with sparse, asymmetric connections show complex dynamical behavior similar to Random Boolean Networks, with a transition from ordered to chaotic dynamics at a critical average connectivity $K_c$. In this type of model - contrary to Boolean Networks - propagation of local perturbations (damage) depends on the in-degree of the sites. $K_c$ is determined analytically, using an annealed approximation, and the results are confirmed by numerical simulations. It is shown that the statistical distributions of damage spreading near the percolation transition obey power-laws, and dynamical correlations between active network clusters become maximal. We investigate the effect of local damage suppression at highly connected nodes for networks with scale-free in-degree distributions. Possible relations of our findings to properties of real-world networks, like robustness and non-trivial degree-distributions, are discussed.