Q:

How do languages with case deal with 'sentential arguments'?

I'm not 100% sure of the proper terminology here, so I'm just using the one used in the LCK.
What I mean by a sentential argument is an argument that is a sentence rather than a noun phrase.
Case in point: 'That people still read that book annoys me'. Here, the 'subject' is an entire sentence complete with its own objects: "(that) people still read that book".
But how do languages that mark nouns for case deal with this? You obviously can't apply a case ending to an entire sentence. Yes, I know some languages, like German, don't mark case on the noun itself, but what about languages that do?

A:

I don't know if Korean case markers qualify as genuine case markers, but you can absolutely add case markers to a whole clause in Korean.  E.g.,

범인이 그 기차를 탔음이 확실하다.  It is certain that the criminal took the train.
[beomin-i [geu gicha]-reul ta-ss-eum]-i hwaksil-ha-da.
[criminal-SUBJ [that train]-OBJ take-PAST-(noun making suffix)]-SUBJ certainty-(verb making suffix)-(sentence ending).

Here the verb of the subordinate clause is turned into a noun form, somewhat like "the criminal having taken the train", and then the subject marker attaches to it.
Or you can add a dummy noun 것 (serving the same role as "that" as in "that people still read that book") and attach the case marker there: think of it as analogous to "The fact that people still read that book annoys me."

