Segmented silicon detectors are widely used in modern high-energy physics (HEP) experiments due to their excellent spatial resolution and well-established manufacturing technology. However, in such experiments the detectors are exposed to high fluences of particle radiation, which causes irreversible crystallographic defects in the silicon material. Since 1990's, considerable amount of research has gone into improving the radiation hardness of silicon detectors. One very promising approach is to use magnetic Czochralski silicon (MCz-Si) that has been found to be more radiation hard against charged hadrons than traditional Float Zone silicon material (Fz-Si) used in the current HEP applications. Other approaches include operating the devices at cryogenic temperatures and designing special detector structures such as p-type detectors or semi-3D detectors. In order to demonstrate that the developed technologies are suitable for the HEP experiments, it is necessary to extensively characterize the potentially radiation hard detectors. We have an excellent instrument for this, the Cryogenic Transient Current Technique (C-TCT) measurement setup, which is an effective research tool for studying heavily irradiated silicon detectors. With the C-TCT setup it is possible to extract the full depletion voltage, effective trapping time, electric field distribution and the sign of the space charge in the silicon bulk in the temperature range of 45-300 K. This article presents a description of this setup and measurement results from detectors processed of MCz-Si.