Data-driven modeling approaches receive significant attention in robotics as they are capable of representing system behavior to which first-order principles cannot be employed. Modeling of human motions, based on observations is one of the many application areas. So far, however, the available probabilistic approaches cannot handle dynamics evolving in the space of rigid motions, as rotations are not appropriately considered. In this article, we present a mathematical framework for Gaussian process modeling, where the valid input domain is generalized to full rigid motions, namely the special Euclidean group SE(3). The kernel functions inside the Gaussian process are modified to exploit properties of the input data representation by dual quaternions. We further prove that the presented covariance functions maintain the Gaussian process properties. The correctness and accuracy of our approach is validated on simulated and real human motion data. We analyze the estimation performance of the novel Gaussian process framework in comparison to state of the art techniques, and show significantly improved model behavior of rigid motions.