A quantum internet, i.e., a global interconnection of quantum devices, is the long-term goal of quantum communications and has so far been based on two-dimensional systems (qubits). Recent years have seen a significant development of high-dimensional quantum systems (qudits). While qudits present higher photon information efficiency and robustness to noise, their use in quantum networks presents experimental challenges due to the impractical resources required in high-dimensional quantum repeaters. Here, we show that such challenges can be met via the use of standard quantum optical resources, such as weak coherent states or weak squeezed states, and linear optics. We report a concrete design and simulations of an entanglement swapping scheme for three-and four-dimensional systems, showing how the network parameters can be tuned to optimize secret key rates and analyzing the enhanced noise robustness at different dimensions. Our work significantly simplifies the implementation of high-dimensional quantum networks, fostering their development with current technology.