(1) Introduction: Dementia is a major public health problem, and Alzheimer’s disease (AD) is the most frequent subtype. Clarifying the potential risk factors is necessary in order to improve dementia-prevention strategies and quality of life. Here, our purpose was to investigate the role of the absence of hedonic tone; anhedonia, understood as the reduction on previous enjoyable daily activities, which occasionally is underdetected and underdiagnosed; and the risk of developing AD in a cognitively unimpaired and non-depressed population sample. (2) Method: We used data from the Zaragoza Dementia and Depression (ZARADEMP) project, a longitudinal epidemiological study on dementia and depression. After excluding subjects with dementia, a sample of 2830 dwellers aged ≥65 years was followed for 4.5 years. The geriatric mental state examination was used to identify cases of anhedonia. AD was diagnosed by a panel of research psychiatrists according to Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition (DSM-IV) criteria. A multivariate survival analysis and Cox proportional hazards regression model were performed, and the analysis was controlled by an analysis for the presence of clinically significant depression. (3) Results: We found a significant association between anhedonia cases and AD risk in the univariate analysis (hazard ratio (HR): 2.37; 95% CI: 1.04–5.40). This association persisted more strongly in the fully adjusted model. (4) Conclusions: Identifying cognitively intact individuals with anhedonia is a priority to implement preventive strategies that could delay the progression of cognitive and functional impairment in subjects at risk of AD.