We developed a generalized spectral phase superposition approach for generating accelerating optical beams along arbitrary trajectories. Such beams can be customized by predefining an appropriate superimposed phase pattern that consists of multiple sub-phases. We generated a spirally accelerating beam in a three-dimensional space and developed an algorithm to improve the uniformity of the intensity along the trajectory by introducing phase-shift factors. We also experimentally verified our numerical simulations. The proposed approach breaks the conventional convex trajectory restrictions. These various accelerating beams would pave the way for optically moving particles along a desired trajectory. The generation of such arbitrary accelerating beams is likely to give rise to new applications in flexible optical manipulation, wave front control, and optical transportation and guidance of particles.