Rydberg-atom quantum simulators are of keen interest because of their possibilities towards high-dimensional qubit architectures. Here we report continuous tuning of quantum Ising Hamiltonians of Rydberg atoms in three-dimensional arrangements. Various connected graphs of Rydberg atoms constructed with vertices and edges respectively representing atoms and Rydberg-blockaded atom pairs, and their eigenenergies are probed along with their geometric intermediates during structural transformations. Conformation spectra of star, complete, cyclic, and diamond graphs are probed for four interacting atoms and antiprism structures for six atoms. The energy level shifts and merges of the tested structural transformations are clearly observed with Fourier-transform spectroscopy, in good agreement with the model few-body quantum Ising Hamiltonian. This result demonstrates the possibility of continuous geometry tuning and thus programming of many-body spin-Hamiltonian systems.