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A nanostrip magnonic-crystal waveguide with spatially periodic width modulation can serve as a gigahertz-range spin-wave filter. Compared with the regular constant-width nanostrip, the periodic width modulation creates forbidden bands (band gaps) at the Brillouin zone boundaries due to the spin-wave reflection by the periodic potential owing to the long-range dipolar interactions. Previous works have shown that there is a critical challenge in tuning the band structures of the magnonic-crystal waveguide once it is fabricated. In this work, using micromagnetic simulations, we show that voltage-controlled magnetic anisotropy can effectively tune the band structures of a ferromagnetic–dielectric heterostructural magnonic-crystal waveguide. A uniformly applied voltage of 0.1 V/nm can lead to a significant frequency shift of ~9 GHz. A spin-wave transistor prototype employing such a kind of spin-wave filter is proposed to realize various logical operations. Our results could be significant for future magnonic computing applications.