Various detection schemes are currently investigated in the terahertz frequency range (<2 THz), as an alternative to the widespread bolometers and Golay cells that feature very high sensitivity but low modulation frequencies (<10 kHz). In this work, we report an alternative concept based on an optomechanical detector able to read out an arbitrary periodic modulation imprinted on a terahertz beam. The detector is a based on a combination of a split-ring resonator, acting as a terahertz antenna, and a high-quality mechanical resonator with typical resonant frequency ω$_0$/2π = 1 MHz. We demonstrate two methods for Fourier sampling the envelope of the terahertz signal, in which the mechanical resonator is used as a reference oscillator. With our methods, signals with an arbitrary period T can be sampled, even those that are much faster than the oscillation period of the mechanical resonator, 1/T ≫ ω$_0$.