Unique combinations of properties such as mechanical compliance and chemical stability make polymers attractive for many applications. However, the intrinsic low thermal conductivity of bulk polymers has generally limited their potential for heat dissipation applications, and in fact they are widely used as thermal insulators. But in recent years, gel-spun, ultra-oriented fibers made of ultra-high molecular weight polyethylene (UHMW-PE) have sparked interest in the thermal management community due to their exceptionally high thermal conductivity. These fibers are typically used in commercially-produced protective gear such as motorcycle jackets and ballistic vests due to their high mechanical strength, but they have not been widely utilized for heat spreading and thermal management applications. While recent studies have characterized individual fibers and ultra-drawn films, the thermal properties of fabrics constructed from these materials remain virtually unexplored. Here, we synthesize plain-weave fabrics from yarns of commercially available gel-spun UHMW-PE and measure the thermal properties of the individual microfibers, yarns, and woven fabrics using an in-house thermal characterization technique based on infrared microscopy. For the woven fabric, we report an effective in-plane thermal conductivity of ~10 Wm^-1K^-1 in the direction aligned with the weft yarns, which is 2-3 orders of magnitude higher than conventional textile materials. This work reveals the high thermal conductivity of UHMW-PE fabrics that can be realized using a scalable textile manufacturing platform and lays the foundation for exploiting their unique thermomechanical properties for heat spreading functions in flexible/wearable devices.