In the last decade, the computerised diagnosis of Alzheimer's disease (AD) and mild cognitive impairment (MCI) using the information provided by different neuroimaging techniques has been extensively studied. However, the texture of such neuroimages has been little explored. In this work, both diagnoses were conducted based solely on the texture of fluorodeoxyglucose-positron emission tomography (FDG-PET) images, which was extracted using a novel 3D extension of the well-known 2D texture descriptor local binary patterns (LBPs). In LBPs, the concepts of uniformity and rotation invariance are of fundamental importance. We show that the proposed approach, unlike other 3D extensions found in the literature, closely replicates these concepts, as originally proposed in the 2D setting. Experimental results showed that the new 3D LBP version is able to enhance the generalisation ability of the diagnostic system and also that the texture of FDG-PET scans contains distinctive information about the presence of both AD and MCI.