We study the X-ray properties of 393 optically selected early-type galaxies (ETGs) over the redshift range of z≈ 0.0-1.2 in the Chandra Deep Fields (CDFs). To measure the average X-ray properties of the ETG population, we use X-ray stacking analyses with a subset of 158 passive ETGs (148 of which were individually undetected in X-ray). This ETG subset was constructed to span the redshift ranges of z= 0.1-1.2 in the ≈4Ms CDF-South and ≈2Ms CDF-North and z= 0.1-0.6 in the ≈250ks Extended-CDF-South where the contribution from individually undetected active galactic nuclei (AGN) is expected to be negligible in our stacking. We find that 55 of the ETGs are detected individually in X-ray, and 12 of these galaxies have properties consistent with being passive hot-gas-dominated systems (i.e. systems not dominated by an X-ray bright AGN). On the basis of our analyses, we find little evolution in the mean 0.5-2keV to B-band luminosity ratio (L _X/L _B∝ [1 +z] ^1.2) since z≈ 1.2, implying that some heating mechanism prevents the gas from cooling in these systems. We consider that feedback from radio-mode AGN activity could be responsible for heating the gas. We select radio AGN in the ETG population using their far-infrared/radio flux ratio. Our radio observations allow us to constrain the duty cycle history of radio AGN activity in our ETG sample. We estimate that if scaling relations between radio and mechanical power hold out to z≈ 1.2 for the ETG population being studied here, the average mechanical power from AGN activity is a factor of ≈1.4-2.6 times larger than the average radiative cooling power from hot gas over the redshift range z≈ 0-1.2. The excess of inferred AGN mechanical power from these ETGs is consistent with that found in the local Universe for similar types of galaxies.