The quality of work organization may be responsible not only for reduced productivity but also for an increased risk of mental and physical disorders. This study was aimed at testing this hypothesis. Workers of a local health unit in Italy were asked to fill out the Work Organization Assessment Questionnaire (WOAQ) during their periodic medical examinations in the second half of 2018. On the same occasion, they also completed the Demand/Control/Support (DCS) measure of job strain, the Effort/Reward Imbalance (ERI) questionnaire, and the General Health Questionnaire (GHQ12) to assess psychological health. A total of 345 workers (85.8%) completed the survey. Linear regression analysis showed that the quality of work organization was inversely proportional to psychological health problems (p < 0.001). Occupational stress, measured both by job strain and ERI, was a moderating factor in this relationship. The relationship between the WOAQ and psychological health, moderated by job strain or ERI, remained highly significant even after adjustment for sex, age, social support, and overcommitment. Regression models explained over 40% of the shared variance of the association between quality of work organization and psychological health. The quality of work organization significantly predicted the risk of sickness absence for musculoskeletal disorders (OR = 0.984, CI95% 0.972–0.996) and for other health problems (OR = 0.977, CI95% 0.967–0.988). A continuous improvement of work organization must consider not only the clients’ or production needs but also the well-being of workers.