Graph learning benefits many fields. However, Graph Neural Networks (GNNs) often struggle with scalability, especially on large graphs. At the same time, many tasks seem to be simple in terms of learning, e.g., simple diffusion yields favorable performance. In this paper, we present Random Propagation GNN (RAP-GNN), a framework that addresses two main research questions: (i) can random propagations in GNNs be as effective as end-to-end optimized GNNs? and (ii) can they reduce the computational burden required by traditional GNNs? Our empirical findings indicate that RAP-GNN reduces training time by up to 58\%, while maintaining strong accuracy for node and graph classification tasks.