We investigate, both theoretically and numerically, a graphene-coated nano-cylinder illuminated by a plane electromagnetic wave in the far-infrared range of frequencies. We have derived an analytical formula that enables fast evaluation of the spectral window with a substantial reduction in scattering efficiency for a sufficiently thin cylinder. This polarization-dependent effect leads to tunable resonant invisibility that can be achieved via modification of graphene chemical potential monitored by the gate voltage. A multi-frequency cloaking mechanism based on dimer coated nanowires is also discussed in detail.