There are several questions here on quoting a question within a sentence, but most of them deal with the quote being at the end of the larger sentence. What if it's in the middle?
In many cultures, "How are you?" is often asked at the beginning of a conversation.
I can't really think of any other way to write this, except maybe to add a comma immediately after the quote, but having a question mark in the middle of a sentence just kind of niggles at me a bit. Is this correct?
That is the correct punctuation. The issue comes up in dialogue as well. If a character asks a question, then does something, it would be written like this: "What's for dinner?" John asked as he strode into the kitchen.
The question mark belongs to his question, and the period after 'kitchen' ends the entire sentence. The only clue (in your case) that the sentence isn't over, is that not only is the next character lower case, but also the clause hasn't even started yet. The intro phrase "In many cultures," is incomplete.
