Active thermal materials like thermal diodes, regulators, and switches have the potential to revolutionize thermal management, creating an opportunity for significant energy savings. We present results on a proposed thermal switching composite that changes its thermal conductivity based on applied strain. The composite is constructed of highly crystalline, high aspect ratio cellulose nanocrystal (CNC) nanorods embedded in a shape-memory polymer matrix. The properties of the matrix allow for changes to the mechanical state to be indefinitely retained and also for the state to be reversed; this work is the first step in demonstrating that the thermal state exhibits similar reversibility. Measurements of the neat matrix polymer show a factor of three increase in thermal conductivity with applied strain up 100% and abrupt decrease beyond this strain level. A twofold increase in the thermal conductivity is achieved for the proof-of-concept composite at 100% strain. By comparing the measured results to a Maxwell mixing model, the primary drivers of the thermal conductivity change are traced to changes in crystallinity of the matrix and CNC alignment.