Global climate change has emerged as a problem in recent years, and its effects will likely continue to increase in the future. Several scientific studies conducted in the Mediterranean region have demonstrated relatively stationary trends for annual precipitation and significant upward trends for mean annual temperature. These trends present several implications, especially in the Greek islands that serve as major summer tourist destinations where the population is already unable to meet their water demands. The aim of this study is to investigate both long- and short-term variations in temperature and precipitation on three Greek islands in the Mediterranean Sea (Mykonos, Naxos, and Kos). The temperature and rainfall trends, as well as their magnitudes at yearly, seasonal, and monthly time steps, were determined using the non-parametric Mann–Kendall trend test. The Standardized Precipitation Index (SPI) was employed to identify the drought periods. According to the results, precipitation slightly increased (almost stationary) in the three islands, although this rise was not statistically significant. All three islands experienced a sharp and statistically significant increase in their mean annual air temperatures. The region may experience drought episodes as a result of the high temperature increase, which would drastically reduce the amount of water, available for use due to the increased evapotranspiration. For the Mediterranean region, the necessity for a drought management strategy to stop or diminish the severity of drought episodes and their effects has grown into a matter of great concern. It is crucial to take measures and conduct relevant research in order to create the conditions for adaptation and mitigation of climate change consequences and the increased appearance of drought phenomena.