Taking your example:
The kids were hungry.
Adding "none of" to the beginning changes the sentence structure.
None of the kids was hungry.
In the first sentence, "kids" is the subject and is clearly plural. Therefore, a plural verb is called for.
In the second sentence, "none" is the subject, and "of the kids" is now an adjective phrase, with "kids" as the object of that phrase and therefore the antecedent of "none". To determine whether a singular or plural verb is needed, it just has to be answered whether "none" is singular or plural.
At least in today's common American vernacular, for this example, neither choice is awkward enough to be considered 'wrong'. One might consider "none" as assuming the same number as its antecedent (making it plural in this example, and context dependent in general), or as simply a contraction of "not one" or "no one" (making it always singular).
Compare with this example, in which using the plural verb would be incorrect:
Not one of the kids was hungry.
I would venture that in speech and informal writing, more people will say "were", but in formal writing, more would write "was". You may get different answers depending on culture and region.