The CUSUM control chart is an effective tool for monitoring both variable and count data. However, it often requires the assumption of known process parameters. To get away from this assumption, the adaptive CUSUM (ACUSUM) procedure has been suggested, with primary focus on the monitoring of variable data. This paper extends the idea of ACUSUM control charts from the monitoring of variable data to the monitoring of count data based on an inhomogeneous Poisson model. The study is motivated by a real example of male thyroid cancer data in New Mexico. The comparison results show that the ACUSUM method generally performs better than the traditional Poisson CUSUM charts and other variations for monitoring a wide range of shifts in the Poisson mean.