Raman micro-spectroscopy is well suited for studying a variety of properties and has been applied to a wide range of areas. Combined with tuneable temperature, Raman spectra can offer even more insights into the properties of materials. However, previous designs of variable temperature Raman microscopes have made it extremely challenging to measure samples with low signal levels due to thermal and positional instabilities as well as low collection efficiencies. Thus contemporary Raman microscope has found limited applicability to probing the subtle physics involved in phase transitions and hysteresis. This paper describes a new design of a closed-cycle, Raman microscope with full polarization rotation. High collection efficiency, thermal stability, and mechanical stability are ensured by both deliberate optical, cryogenic, and mechanical design. Measurements on two samples, Bi2Se3 and V2O3, which are challenging due to low thermal conductivities, low signal levels, and/or hysteretic effects, are measured with previously undemonstrated temperature resolution.