If one adoopts the Pesetsky's conjecture about the nature of Case, according to which Case is an uninterpretable Tense feature and CP (sentential argument) is the same category as DP - C/DP, complementizers like "that" carry the Case feature in themselves, so it will be enough to only make use of the complementizer to mark the Case. All languages I know stick with this strategy.
Another issue is how to tackle sentential arguments when they come in prepositional objects. Gerund phrases take care of it in English (e.g. "She doesn't believe in getting lost in the wood"). Nevertheless, other languages don't like non-finite complementation to that extend. In Russian, for example, there are so-called pronominal-correlative sentences: the subordinate clause is finite, in its head is a complementizer, the predicate in the higher clause requires a preposition, the preposition assignes a particular case to its object, the complementizer of the lower clause cannot take a case value, so an expletive (demonstrative pronoun in this case) is to be made use of in order to receive a case:
"Ona ne verit v to, čto zabludilaś v lesu"
[She not believe-Pres-3pSg in that (dem.pron.)-AccSg, that (complementizer) get lost-Past-Fem in (the) woods-Preposition.CaseSg].
If there is no expletive, one can simply get rid of the preposition:
"Ona ne verit, čto zabludilaś v lesu".
I am not sure whether this construction is grammatical in English ("She doesn't believe that she has got lost in the woods").
When the complementizer is not "that" (čto), the situation remains the same:
"Ona rasskazala o tom, kak provela leto"
[She tell-Past-Fem about that (dem.pron)-Preposition.CaseSg, how (complementizer) spend-Past-Fem (the) summer-AccSg]
"She told about how she spent the summer".
Once again, the preposition can be ommitted.
Returning to your example 'That people still read that book annoys me' - the Russian translation will be with the expletive too, I mean, not simply "that people read", but "That (dem.pron)-Nom, that people read that book annoys me". Nonetheless, when the sentential argument is not in the subject position, the expletive is not necessary.
The conclusion is that in Russian the case value on sentential arguments is absent, except for when they appear as the arguments of PPs or in the subject position: then the case values are assigned to the expletive.
I guess, this expletive pronoun can be substituted by "the fact", so in Russian "fact" dropped and only "the" (which I indicate by "that") is present. With such an analysis the English constructions are essentially parallel to the Russian ones.