I don’t hear this label used as much as verse-by-verse preaching, but I have come across it. On the surface, again, it sounds like it fits right in with a high view of Scripture and an expository view of preaching. But again, I think it could lead preachers into some unhelpful practices that aren’t truly expository.
Some thoughts to chew on:
1. Every word is a doorway to endless digressions, but how they are working together is the real issue. If you have read the Bible or anything vaguely theological, or if you own a concordance, then any word can be the start of a digression. There are times when cross-referencing is helpful, but it has to be helpful to something specific. That is, it cannot be an end in itself. So you have to study the whole passage in order to know what the main idea is, and then determine whether pursuing the thoughts a word might spark would actually help the communication of that main idea.
2. Every word can slow you down, but you need to give the message of the passage. This follows on from number 1. Since each word, or even each theologically weighty word, can be the start of a digression in your explanation, it follows that every word can slow you down in preaching. For instance, I can imagine someone preaching Ephesians 1:15-23 in this way and running out of time before really preaching the heart of the passage in verses 18-19, or the elaboration of the final element in the subsequent verses. Just because verses 15-17 contain some mighty terms doesn’t change the fact that Paul is really introducing at that point. Preach them, but don’t miss the message of the passage.
3. Every word should be studied, but not every word should be “preached.” Some preachers feel it is their duty to offer a concise word study and chain reference guide for every term in the passage that seems weighty. Study them all. Preach the passage.
4. Every last word is inspired, but they are not equally weighted. I would hold to a verbal plenary inspiration position. That is, God inspired the words (verbum), all of them (plenary). That doesn’t mean that I treat them as individual data banks and thought units when preaching. In any sentence, the meaning is conveyed by a combination of the potential meaning of each word, determined by its function and role within the flow of the sentence and broader context.
Study the words, study the details, do the work. And the work includes the integration of that study. What is the author trying to communicate? What are you trying to communicate to your listeners? Then, preach the passage.
Biblical Preaching 
I think this is a helpful perspective on the idea of “preaching every word.” There does seem to be a danger in that phrase because it can imply that we need to provide an exhaustive word study of every word in the passage. Words derive meaning from context and flow of thought. The danger in studying every word is “illegitimate totality transfer” – that is, taking the meanings of the word elsewhere in Scripture and then loading all of them all up onto the occurrence of that word in the passage being preached. As D.A. Carson says, this causes the preacher to build a systematic theology on every word instead of analyzing the logical flow of the passage and using that to determine the significance of words.
Perhaps it is better to say that expository preaching should be “thought by thought” rather than “word by word.” The point is to convey the thoughts in the passage, and words should be analyzed and broken down only to the extent necessary to convey the thoughts the passage contains.