The most widely-studied mechanism of mass loss from extrasolar planets is photoevaporation via XUV ionization, primarily in the context of highly irradiated planets. However, the EUV dissociation of hydrogen molecules can also theoretically drive atmospheric evaporation on low-mass planets. For temperate planets such as the early Earth, impact erosion is expected to dominate in the traditional planetesimal accretion model, but it would be greatly reduced in pebble accretion scenarios, allowing other mass loss processes to be major contributors. We apply the same prescription for photoionization to this photodissociation mechanism and compare it to an analysis of other possible sources of mass loss in pebble accretion scenarios. We find that there is not a clear path to evaporating the primordial atmosphere accreted by an early Earth analog in a pebble accretion scenario. Impact erosion could remove ~2,300 bars of hydrogen if 1% of the planet's mass is accreted as planetesimals, while the combined photoevaporation processes could evaporate ~750 bars of hydrogen. Photodissociation is likely a subdominant, but significant component of mass loss. Similar results apply to super-Earths and mini-Neptunes. This mechanism could also preferentially remove hydrogen from a planet's primordial atmosphere, thereby leaving a larger abundance of primordial water compared to standard dry formation models. We discuss the implications of these results for models of rocky planet formation including Earth's formation and the possible application of this analysis to mass loss from observed exoplanets.