Motivated by the wide-ranging applications of Hamiltonian decompositions in distributed computing, coded caching, routing, resource allocation, load balancing, and fault tolerance, our work presents a comprehensive design for Hamiltonian decompositions of complete $k$-uniform hypergraphs $K_n^k$. Building upon the resolution of the long-standing conjecture of the existence of Hamiltonian decompositions of complete hypergraphs, a problem that was resolved using existence-based methods, our contribution goes beyond the previous explicit designs, which were confined to the specific cases of $k=2$ and $k=3$, by providing explicit designs for all $k$ and $n$ prime, allowing for a broad applicability of Hamiltonian decompositions in various settings.