This article presents an approach to forecasting count time series with a form of exponential smoothing built from observation‐driven models. The proposed method is easy to implement and simple to interpret. A variant of the approach is also proposed to handle the impact of outliers on the forecast. The performance of the methodology is studied with simulations and illustrated with an analysis of the number of monthly cases of dengue fever observed in Italy for the years 2008–2021. An R package is made available to enable the reader to reproduce the results discussed in the article.