We report a new technique for torsional testing of materials under giga-pascal pressures, which uses a shearing module in a large-volume Paris-Edinburgh press in combination with high-resolution fast radiographic x-ray imaging. The measurement of the relative amplitude and phase lag between the cyclic displacement in the sample and a standard material (Al2O3) provides the effective shear modulus and attenuation factor for the sample. The system can operate in the 0.001-0.01 Hz frequency range and up to 5 GPa and 2000 K although high-temperature measurements may be affected by grain growth and plastic strain. Preliminary experimental results on San Carlos olivine are in quantitative agreement with previously reported Q-1 factors at lower pressure. This cyclic torsional loading method opens new directions to quantify the viscoelastic properties of minerals/rocks at seismic frequencies and under pressure-temperature conditions relevant to the Earth's mantle for a better interpretation of seismological data.