Hurst's rescaled range analysis is a useful tool in the examination of a time series and is designed to measure memory content and determine its fractal texture. This study applies the Hurst method to a new earthquake catalogue for Greece. The study also adopts Monte Carlo simulations to provide a statistical test underpinning the Hurst analyses. Together these reveal basic temporal fractal characteristics in the earthquake occurrence time-histories' memory. Three regions are considered, approximately: all of Greece and some surrounding areas, and the sub-zones of the Hellenic Arc and the Gulf of Corinth. Three temporal textures are considered: elapsed time between earthquakes, strain energy release, and earthquake frequency. The elapsed temporal textures for the zone whole Greece indicate distinct characteristics in chronological order and possess long memory. These belong to the class non-random pattern. However, these characteristics generally disappear when the sub-zones are considered and become random patterns. The Monte Carlo simulations support this. Therefore, memoryless statistical seismic hazard estimates may not be suitable for whole Greece but could be useful for the sub-zones. The strain energy release temporal textures for whole Greece and for the sub-zones, no matter that these seem to possess long memory at first analysis, are all random patterns. In other words, the Monte Carlo simulations demonstrate that these patterns are much more likely to happen by chance. The seismic frequency textures for whole Greece and for the sub-zones suggest long memory, however, only the texture for the Hellenic Arc zone (MS ≥ 5.0) and that for whole Greece (MS ≥ 4.0) approach demonstrable non-random patterns. Except for these, other patterns happen by chance.