I've always been taught to use "that" to introduce a subordinate clause only if there is an ambiguity in the meaning of a sentence without it.
He said on Monday he had lost his dog. (unclear)
He said that on Monday he had lost his dog. (clear)
He said on Monday that he had lost his dog. (clear)
In sentences where it is not necessary to clear up any ambiguity, I've been taught to always drop "that".
He said he is going to be late.
"Rasoul said he is confident the military will retake other cities captured by ISIS." CNN.com
But I have seen instances where writers use that but it doesn't help clarify the sentence.
"Warren said that it doesn't appear that ISIS has the strength to take Ramadi back." CNN.com
Does using "that" to introduce a subordinate clause serve any other purpose but to eliminate ambiguity?