We propose a new multiple quasi-phase-matched wavelength converter based on the continuous phase modulation of a χ(2) grating for use in variable wavelength conversion. A numerical study shows that the proposed device exhibits a high conversion efficiency, flexible design, and robust fabrication tolerance. A waveguide device fabricated by annealed proton exchange agrees well with the numerical design. Fine-tuning the device enabled us to demonstrate variable wavelength conversion between signals on the standard optical frequency grid. Using the device, we also demonstrated fast (<100 ps) wavelength switching of 4-channel 40-Gb/s signals. The obtained results clearly show that the proposed multiple quasi-phase-matched devices will be useful when constructing future flexible photonic networks.