Modelling based on the Birnbaum-Saunders distribution has received considerable attention in recent years. In this article, we introduce a new approach for Birnbaum-Saunders regression models, which allows us to analyze data in their original scale and to model non-constant variance. In addition, we propose four types of residuals for these models and conduct a simulation study to establish which of them has a better performance. Moreover, we develop methods of local influence by calculating the normal curvatures under different perturbation schemes. Finally, we perform a statistical analysis with real data by using the approach proposed in the article. This analysis shows the potentiality of our proposal.