We present results of the search for variable stars in a sample of Local Group isolated dwarf galaxies, namely LGS3, IC1613, Cetus, and Tucana, based on very deep HST/ACS data. About 900 variables and candidates have been found in total. Among the confirmed variables, we identified about 700 RR Lyrae stars, 60 Cepheids, and 50 eclipsing binaries from their light-curve morphology and location in the color-magnitude diagrams. We find that the properties of these variables reflect details of the star formation history of the galaxies. In particular, our in depth study of the RR Lyrae population of the dSph Tucana reveals interesting details about the early evolution of this galaxy, and the origin of its stellar population gradients. We find that the fainter RR Lyrae stars, also having a shorter period, are more centrally concentrated than the more luminous, longer period RR Lyrae variables. Through comparison with the predictions of theoretical models of stellar evolution and stellar pulsation, we interpret the fainter RR Lyrae stars as a more metal-rich subsample, suggesting that a metallicity gradient must have appeared very early on in the history of this galaxy. We also compare the location of the Cepheids found in the galaxies of our sample in the period-luminosity diagram with the Magellanic Clouds Cepheids of the OGLE collaboration, and find that the Cepheids in the gas-rich dwarf galaxies are classical Cepheids, while Cetus and Tucana harbor only anomalous Cepheids.