Nanotechnology is enabling the development of miniature devices able to perform simple tasks at the nanoscale. The interconnection of such nano-devices with traditional wire- less networks and ultimately the Internet enables a new network- ing paradigm known as the Internet of Nano-Things (IoNT). De- spite their promising applications, nano-devices have constrained power, energy and computation capabilities along with very limited memory on board, which may only be able to hold one packet at once and, thus, requires packets to be delivered before certain hard deadlines. Towards this goal, a fully-distributed computation-light provably-correct scheduling/MAC protocol is introduced for bufferless nano-devices, which can maximize the network throughput, while achieving perpetual operation. More specifically, the proposed scheduling algorithm allows every nano- device to make optimal transmission decisions locally based on its incoming traffic rate, virtual debts, and channel sensing results. It is proven that the proposed algorithm is timely- throughput optimal in the sense that it can guarantee reliable data delivery before deadlines as long as the incoming traffic rates are within the derived maximum network capacity region. This feature not only can lead to high network throughput for the IoNT, but also guarantees that the memory of each device is empty before the next packet arrives, thus addressing the fundamental challenge imposed by the extremely limited memory of nano-devices. In addition, the optimal deadline is derived, which guarantees that all the nano-devices can achieve perpetual communications by jointly considering the energy consumption of communications over the THz channel and energy harvesting based on piezoelectric nano-generators