I just noticed that one of the polls on the main page for the LOTR Wikia is 'Which movie do you think stayed closest to the book?' or something like that. To my honest surprise, Fellowship of the Ring got top billing (by a freaking mile!). I personally think that it is probably the furthest from the books, and here are my reasons.
The first is that a hell of a lot of parts are missing. The Old Forest, Tom Bombadil, Bill Ferny, Buckland, Glorfindel and a bunch more. The Two Towers kept a lot of scenes, even if it was done on a slightly different timeline. Meanwhile, the Fellowship movie glances over the fact that Pippin travelled with Frodo and Sam from Bag End, Frodo stayed in the Shire for nearly two decades after Bilbo's party and the complete lack of Fatty Bolger, Crickhollow, Old Man Willow and Tom Bombadil. It could be argued that the last two take away from the story, but I disagree. Firstly, it's in the Barrow-downs that Merry gets the dagger he later used to kill the Witch-King, plus Tom Bombadil is great breakaway from all the grim stuff that the Hobbits had to put up with (the Nazgul, having to wak across the Shire, etc.).
The second is that there are a lot of awesome and tense moments in the book that Peter Jackson ignored, the main one in my opinion being the Wolf attack on Caradhras. That was an epic moment, and a daring one for the Fellowship, and it really added a lot of tension to the journey. Some more moments include when Haldir took Frodo up through the trees to look over Caras Galadhon. That would have been amazing to see in the movies and it would have added a bit more character to Haldir, who was incredibly bland in the movies. Say what you will, but some of the moments that Peter Jackson left out are some of the best in the book.
My final point is that the Two Towers (the one with the least votes in the poll) probably stayed most true to the books. The only reasons to be argued are that the movie finishes earlier than the books and in the movie trilogy, Saruman is killed instead of him taking over the Shire. Other than these, the books and movies are parallel, with the same characters, similar dialogues and the same scenes in the same order, even though the movie started later and finished earlier than the book. I guess you could also mention other small points, such as the fact that Faramir seems very different in the movies. In the books, he resists the Ring instantly and is probably the most heroic hero of all the heroes, while in the movies he is at first tempted, but eventually resists the Ring and realizes that it would only bring doom to Gondor, not save it. However, the other characters are the same and the only scenes left out are nowhere as major as those left out in Fellowship.
That's my little rant on why I think the poll is wrong, and I would love to hear some feedback (whether you agree, or disagree, I don't care).