Bitumen and high-quality subangular aggregates, the two principal materials used for asphalt concrete construction, are finite and expensive materials. The general availability of crumb rubber and naturally occurring aggregates of different shapes, especially flat and elongated shapes, indicates that they are feasible alternative materials for expanding the volume of bitumen and utilizing a wider range of aggregate shapes for the development of asphalt concrete, with an associated environmental benefit. The study investigated the effect of adding up to 15% crumb rubber and aggregates sorted into different groups, i.e., rounded, elongated, flat, and their combinations, on the rheological and mechanical properties and durability of 50/70 of hot-mix asphalt pavement. The addition of crumb rubber decreased ductility and penetration but increased the softening point. For a 5.5% bitumen content, asphalt concrete briquettes consisting of 7% crumb rubber and three types of aggregate shapes, i.e., 100% rounded, a mix of 75% rounded and 25% elongated, and a mix of 75% rounded, 15% elongated and 10% flat, were associated with high Marshall stability and indirect tensile strength as well as low lateral deformation due to their high solidity and moderate angularity ratio. Also, the addition of 7% crumb rubber resulted in a significant improvement in the tensile strength ratio and rebound strain of briquettes consisting of 75% rounded and 25% elongated aggregates and those with 75% rounded, 15% elongated and 10% flat aggregates. In relation to the parameters investigated, the three groups of briquettes met some of the local (South Africa) requirements for the surface course and base course of low traffic volume roads.