We report the results of a long BeppoSAX observation of Abell 3667, one of the most spectacular galaxy cluster in the southern sky. A clear detection of hard X-ray radiation up to ~ 35 keV is reported, while a hard excess above the thermal gas emission is present at a marginal level that should be considered as an upper limit to the presence of nonthermal radiation. The strong hard excesses reported by BeppoSAX in Coma and A2256 and the only marginal detection of nonthermal emission in A3667 can be explained in the framework of the inverse Compton model. We argue that the nonthermal X-ray detections in the PDS energy range are related to the radio index structure of halos and relics present in the observed clusters of galaxie.