Yes, the structure is perfectly correct and logical - why on earth did you think it is incorrect? As other members have pointed out, "say" needs an object to make the sentence complete ("this great country" is not the object; it is where the people are from), and the structure sounds like it's from a formal speech.

It is incomplete, because "say" in this case needs to take an object. So you need more to this sentence (for instance, "I say to all the people of this great country: we must not despair." This particular construction is rare; it's usually found in political speeches.

As it stands, the sentence is awkward because it begs the question "WHAT do you say to the people of this great country?" It does sound like something out of a politician's campaign speech or a statesman's speech in a foreign country. It would make more sense if the sentence was something like: I say to all the people of this great country, "It's morning again in America. We have suffered through a recession but, as your next President, we will once again become a strong economy and regain our position as leader of the Free World." (something Ronald Reagan might have said while campaigning for President in 1980) Or: I say to all the people of this great country, "Your destiny is in your hands. America stands ready to help but first, you must resolve your sectarian differences and come together in unity to defend your nation." (something Barack Obama might say to the Iraqis)

More often, you would "tell someone about something". Or "tell somebody about something." Typically in English the verb "tell" has a direct object pronoun, and say does not. However, you would also be understood by native speakers if you said "say to somebody of something", it's just not as grammatically correct and sounds a bit awkward. Hope that helps.

"I say to all the people of this great country" is an incomplete sentence, it's only half of the statement. Therefore, the structure that you are suggesting does not seem to work. With Sean's example, THEN the sentence is complete and it does follow the structure. This is a very formal structure and is not used in conversation, or even professionally, but in speeches.