This paper proposes a new methodology for the interpolation of a given set of 3D rotation poses that have to be reached in successive times by preserving continuity in orientation, angular velocity and angular acceleration. The discussed algorithm ensures the generation of smooth angular trajectories without singularities. The distinctive features of the proposed approach are the straightforward formulation, the reduced computational burden and the lack of iterative procedures. The presented methodology has applications in the generation of spatial motion of mechanical systems (e.g. robotics, flying devices) or in 3D computer graphics. After a theoretical introduction, the proposed algorithm is compared with other methods available in literature and some possible applications are presented.