The first time I read LotR, I read it straight through, because I felt all excited about reading it and felt the need to be proper. And I swear, I never skipped a page, even during the Council of Elrond!
Once when I was sick I started in the middle of
The Two Towers and read only to the end of the Battle of Helm's Deep. I must say it was a very enlightening read, as it had been some time since I'd read any of the books and it was very absorbing. 'Tis sad that I never seem to read as well as when I'm sick, as I feel like doing nothing else.
I'm pretty sure you can pick up
The Lord of the Rings at any poin in the story and just read. Any part of the story is just as good as any other part. It may seem sacriligious, but...why not?
Often when I pick any of the books up just to reference something, I start reading and can't stop!