Oscillators, which produce continuous periodic signals from direct current power, are central to modern communications systems, with versatile applications such as timing references and frequency modulators. However, conventional oscillators typically consist of macroscopic mechanical resonators such as quartz crystals, which require excessive off-chip space. Here we report oscillators built on micron-size, atomically-thin graphene nanomechanical resonators, whose frequencies can be electrostatically tuned by as much as 14%. The self-sustaining mechanical motion of the oscillators is generated and transduced at room temperature by simple electrical circuitry. The prototype graphene voltage controlled oscillators exhibit frequency stability and modulation bandwidth sufficient for modulation of radio-frequency carrier signals. As a demonstration, we employ a graphene oscillator as the active element for frequency modulated signal generation, and achieve efficient audio signal transmission.