We study the X-ray emission in a sample of galaxy clusters using the BeppoSAX PDS instrument in the 20 – 80 keV energy band . We estimate the non-thermal hard X-ray cluster emission ( HXR ) by modeling the thermal contribution from the cluster gas and the non-thermal contamination from the unobscured AGN in the clusters . We also evaluate the systematic uncertainties due to the background fluctuations . Assuming negligible contamination from the obscured AGN , the resulting non-thermal component is detected at a 2 \sigma level in \sim 50 % of the non-significantly AGN-contaminated clusters : A2142 , A2199 , A2256 , A3376 , Coma , Ophiuchus and Virgo . The data are consistent with a scenario whereby relaxed clusters have no hard X-ray component of non-thermal origin , whereas merger clusters do , with a 20 – 80 keV luminosity of \sim 10 ^ { 43 - 44 } h _ { 50 } ^ { -2 } erg s ^ { -1 } . The co-added spectrum of the above clusters indicates a power-law spectrum for the HXR with a photon index of 2.8 ^ { +0.3 } _ { -0.4 } in the 12 – 115 keV band , and we find indication that it has extended distribution . These indications argue against significant contamination from obscured AGN , which have harder spectra and centrally concentrated distribution . These results are supportive of the assumption of the merger shock acceleration of electrons in clusters , which has been proposed as a possible origin of the non-thermal hard X-ray emission models . Assuming that the Cosmic Microwave Background photons experience Inverse Compton scattering from the merger-accelerated relativistic electrons , and thus produce the observed HXR , the measured hard X-ray slope corresponds to a differential momentum spectra of the relativistic electrons with a slope of \mu = 3.8 – 5.0 . In presence of cluster magnetic fields this relativistic electron population produces synchrotron emission with a spectral index of 1.4 – 2.1 , consistent with radio halo observations of merger clusters . Thus both hard X-ray and radio observations of merger clusters are consistent with the Inverse Compton model . The observed slope of HXR is also consistent with that predicted by the non-thermal bremsstrahlung , which thus can not be ruled by the fit to the current data , even though this model requires an extreme , untenable cluster energetics . Assuming centrally concentrated distribution of HXR , the data requires a harder slope for the HXR spectrum , which is consistent with secondary electron models , but this model yields a worse fit to the PDS data and thus seems to be disfavored over the primary electron Inverse Compton model .