Jerusalem artichoke (Helianthus tuberosus L.) is a valuable source of inulin and fructooligosaccharides—compounds with well-documented prebiotic and functional food properties. However, its high moisture content significantly limits storage stability. This study aimed to assess the effects of drying method and process temperature on the drying kinetics and selected physicochemical properties of Jerusalem artichoke. Convective drying (AD) and combined convective–microwave drying (AMD), using a microwave power of 100 W, were employed. Drying was conducted at air temperatures of 40 °C, 60 °C, and 80 °C. Among the mathematical models evaluated, the Page model provided the best fit to the experimental drying data for both methods. Samples dried at 80 °C using the AMD technique exhibited the most pronounced changes in color, significant polyphenol losses, and a substantial reduction in antioxidant capacity compared to the fresh material. The lowest polyphenol degradation and the highest retention were observed in products dried at 40 °C using both AD and AMD methods. Notably, the AMD method significantly reduced drying time and improved the grindability of the dried Jerusalem artichoke samples. Although AMD contributed to certain quality deterioration, it also promoted a higher degree of particle size reduction. However, this increased degree of particle size reduction had only a limited effect on the extraction efficiency of fructooligosaccharides and inulin. The results of the present study suggest that AMD may serve as a competitive alternative to AD for drying Jerusalem artichoke, particularly when processing time and grindability are critical considerations.