We demonstrate an approach to achieve optically tunable microwave frequency downconversion based on an optoelectronic oscillator (OEO) incorporating a tunable microwave photonic filter. The wideband tunable local oscillation (LO) is generated in the OEO through simply tuning the frequency difference between the optical carrier and the reflection notch of a phase-shifted fiber Bragg grating (PS-FBG). The LO and the input radio-frequency (RF) signal are combined and added to the OEO loop by a single-phase modulator. Through transmitting one modulation sideband of the LO via the reflection notch of the PS-FBG and combining it with the optical carrier split from the laser source, the oscillation of the LO in the OEO is maintained. The reflected modulation sidebands of the LO and the RF signal from the PS-FBG are exported out of the OEO loop and enter a narrowband photodetector to achieve optically tunable microwave frequency downconversion. Our method is experimentally evaluated, in which optically tunable LOs in the frequency range 6-15 GHz are generated, and RF signals in the frequency range 7-16 GHz are successfully downconverted to intermediate frequency band around 1 GHz.