Nowadays, a vast amount of georeferenced data pertains to human and natural activities occurring in complex network-constrained regions, such as road or river networks. In this article, our research focuses on spatio-temporal point patterns evolving over time on linear networks, which we model as inhomogeneous Poisson point processes. Within this framework, we propose an innovative nonparametric method for intensity estimation that leverages penalized maximum likelihood with roughness penalties based on differential operators applied across space and time. We provide an efficient implementation of the proposed method, relying on advanced computational and numerical techniques that involve finite element discretizations on linear networks. We validate the method through simulation studies conducted across various scenarios, evaluating its performance compared to state-of-the-art competitors. Finally, we illustrate the method through an application to road accident data recorded in the municipality of Bergamo, Italy, during the years 2017–2019.