High-index-contrast optical waveguides are crucial for the development of photonic integrated circuits with complex functionalities. Despite many similarities between optical and acoustic waves, high-acoustic-index-contrast phononic waveguides remain elusive, preventing intricate manipulation of phonons on par with its photonic counterpart. Here, we present the realization of such phononic waveguides and the formation of phononic integrated circuits through exploiting a gallium-nitride-on-sapphire platform, which provides strong confinement and control of phonons. By demonstrating key building blocks analogous to photonic circuit components, we establish the functionality and scalability of the phononic circuits. Moreover, the unidirectional excitation of propagating phononic modes allows the exploration of unconventional spin-orbit interaction of phonons in this circuit platform, which opens up the possibility of novel applications such as acoustic gyroscopic and non-reciprocal devices. Such phononic integrated circuits could provide an invaluable resource for both classical and quantum information processing.