There are many who advocate a verse-by-verse approach to preaching. Some entire denominations take this approach. Some (wrongly) define expository preaching by this form. Here are Steve Mathewson’s lists of strengths and weaknesses of the approach.
Strengths. 1. Verse-by-verse sermons dig deeply into the text, thereby countering the contemporary trend toward biblical illiteracy. 2. Verse-by-verse sermons lead the preacher to follow the contours of the text rather than an artificial outline. 3. Verse-by-verse preaching has a tendency to real the author’s intent rather than imposing an idea onto the text.
Weaknesses. 1. The verse-by-verse approach does not serve all literary genres of Scripture equally well. 2. The verse-by-verse approach sometimes results in sermons that lack unity, wherein there is much analysis, but little synthesis. It is possible to obscure the flow of thought in a text by giving emphasis to every passing detail. 3. There is a tendency in verse-by-verse preaching to overload the sermon with raw data and short-change application. 4. Verse-by-verse preaching can slow the preacher’s pace so much that a congregation does not get to hear the whole counsel of God over a reasonable period of time.
(See Mathewson’s chapter 110 in The Art and Craft of Biblical Preaching, pp407ff).
Biblical Preaching 
H. A. “Harry” Ironside excelled in verse-by-verse exposition. I’m sure his popular books (still in print) have been a model for countless others.
After a brief affiliation with the Salvation Army where he learned to do evangelistic preaching, he returned to the Plymouth Brethren, the fellowship of his parents and his own early childhood. This was the church of such expositors as J. N. Darby, William Kelly, and C. H. Macintosh. The influence of their tradition of verse-by-verse, chapter-by-chapter preaching through whole books of the Bible is plain in Ironside’s expositions.
For more than three decades, he traveled the U.S. and Canada in Bible Conference preaching and teaching. During a dozen of those years he also devoted two months each summer to mission work among the Indians of the American Southwest. Though his own formal education ended with eighth grade, he held many meetings under the direction of the Moody Bible Institute and was a frequent visiting professor in the early years of Dallas Theological Seminary. His only pastorate was eighteen years at Moody Memorial Church in Chicago. He was often away from that pulpit on preaching tours until he resigned in 1948 to continue traveling the US, Canada, the British Isles and elsewhere. He died on a preaching tour in New Zealand in 1951. — abt