I've just been catching up with Corey Olsen's exploration of Lord of the Rings, and he makes the same point regarding Tolkien: the man thought in alliterative prose, and there are some places where it stands out even to the eye, but you can catch a lot more of the alliteration by reading passages out loud.

The example that brought it up was towards the end of the attempt to climb the Redhorn: "Soon they all stood once more on the flat shelf at the head of the steep slope where they had felt the first flakes of snow the night before." (The F in 'before' might not line up with the previous F's unless you've been reading out loud.)