Super-Bloch oscillations (SBOs) offer a powerful route to control wave packet propagation in driven lattices, yet their realization in magnonic systems has remained elusive. Here, we predict SBOs of magnon wave packets in periodic magnonic crystals driven by static and ac gradients of magnetic anisotropy field. A static gradient alone induces magnon Bloch oscillation (BO) with a constant carrier frequency. Remarkably, adding a near-resonant MHz-scale ac driving gives rise to a slow large-amplitude envelope oscillation, the hallmark of SBO. At exact resonance, the ac driving enables phase-controlled directional transport with a carrier-frequency shift up to GHz scale. Consequently, the same structure can be switched among three distinct regimes: conventional BO, SBO, and directed transport. Analytical expressions for the oscillation amplitudes, superperiod, drift velocity, and frequency evolution are validated by micromagnetic simulations. Extending to antiferromagnet magnonic crystals, we further demonstrate that these effects drive opposite motions of two oppositely polarized magnon modes in THz range. Our work establishes magnon SBOs as a versatile platform, opening a new avenue for reconfigurable magnonic devices.