Summary: Numerical generalized randomized Hamiltonian Monte Carlo is introduced, as a robust, easy to use and computationally fast alternative to conventional Markov chain Monte Carlo methods for continuous target distributions. A wide class of piecewise deterministic Markov processes generalizing Randomized HMC (Bou-Rabee and Sanz-Serna) by allowing for state-dependent event rates is defined. Under very mild restrictions, such processes will have the desired target distribution as an invariant distribution. Second, the numerical implementation of such processes, based on adaptive numerical integration of second order ordinary differential equations (ODEs) is considered. The numerical implementation yields an approximate, yet highly robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of the complete Hamiltonian trajectories (hence, the title). The proposed algorithm may yield large speedups and improvements in stability relative to relevant benchmarks, while incurring numerical biases that are negligible relative to the overall Monte Carlo errors. Granted access to a high-quality ODE code, the proposed methodology is both easy to implement and use, even for highly challenging and high-dimensional target distributions. Supplementary materials for this article are available online.