Interpretability has become a necessary feature for machine learning models deployed in critical scenarios, e.g. legal system, healthcare. In these situations, algorithmic decisions may have (potentially negative) long-lasting effects on the end-user affected by the decision. In many cases, the representational power of deep learning models is not needed, therefore simple and interpretable models (e.g. linear models) should be preferred. However, in high-dimensional and/or complex domains (e.g. computer vision), the universal approximation capabilities of neural networks are required. Inspired by linear models and the Kolmogorov-Arnold representation theorem, we propose a novel class of structurally-constrained neural networks, which we call FLANs (Feature-wise Latent Additive Networks). Crucially, FLANs process each input feature separately, computing for each of them a representation in a common latent space. These feature-wise latent representations are then simply summed, and the aggregated representation is used for prediction. These constraints (which are at the core of the interpretability of linear models) allow a user to estimate the effect of each individual feature independently from the others, enhancing interpretability. In a set of experiments across different domains, we show how without compromising excessively the test performance, the structural constraints proposed in FLANs indeed facilitates the interpretability of deep learning models. We quantitatively compare FLANs interpretability to post-hoc methods using recently introduced metrics, discussing the advantages of natively interpretable models over a post-hoc analysis.