The "diagonal method" appears (as seen on a site dedicated to its advocacy) to have been invented — he says "discovered" — in 2006 by photographer and photography teacher Edwin Westhoff. The "method" is simple. It states that details which are important to the artist will be found — within a very close precision — along an imaginary diagonal line drawn at 45° from one one of the corners.
The rule states that these important details will be no more than 1.5mm from the diagonal on an A4 print. That's about 0.5%, which is the width of the red lines in the illustration I've made. So, in order to fit the diagonal method, a picture must have details precisely on one of those red lines; if it doesn't the diagonal method does not apply. The rule makes no claims about where on the lines the details should fall.
No particularly strong reason is given for the importance of these lines; Westhoff writes that this is a discovery based on observation. He suggests that it might be because of the way artist and viewer's eyes track over an image. There's no analysis given of these particular 45% lines over, for example, lines going corner-to-corner, or to divisions of the golden ratio, or otherwise.
Westhoff makes two arguments in favor of the diagonal method over the rule of thirds. First, he says that because this rule requires more precision than the approximation-is-okay statements that usually accompany the rule of thirds, it is more testable. Second, he argues that it is not a rule for overall composition, but an indicator of "details which are important to the artist in a psychological or emotional way." Notably, this importance could even be subconscious.
The first point seems somewhat disingenuous. An article on the above web site "testing" the method starts first by discarding all photos where the author judges the diagonals not to be meaningful; then, of the remaining ones, some fraction have some details determined to fit the method. This is textbook confirmation bias, so color me a bit skeptical on that.
And the second, well... it's decidedly subjective. Without artist's statements, it's hard to be certain whether the chosen details really are the ones of particular importance, or if Westhoff's analysis is simply circular. The idea that the "method" may operate at a subconscious level even beyond the artists' perception is even more untestable — perhaps thousands of people could be asked to identify the "important details" of a large random selection of images, and the aggregate results compared against the predicted position on the diagonal. But no such study has been done.
But on secret knowledge of artists' intent, the "method" is in very good company, because the earlier rules make similar claims. Actually, they make similar claims on both of these counts.
Rabatment of the Rectangle is a quite similar concept; in fact, Westhoff comes near to describing it in his description of his rule: "I called this the Diagonal Method because these lines are also the mathematical diagonals of the two overlapping squares within a rectangle." The imaginary lines forming those two squares are said, by Charles Bouleau in The Painter's Secret Geometry in 1963, to be found throughout history in painting — again, perhaps subconsciously.
The Rule of Thirds appears to have been invented by John Thomas Smith in around 1797. The interesting thing here is that Smith doesn't seem to go for the imprecision Westhoff dismisses his rule for. He seems pretty certain that this ratio is precisely the best way to divide lines or areas. Over the centuries, of course, the rule in practical application hasn't held up to that sort of exactness, and in fact is really quite useful when applied generally and not as a fixed rule. But overall, Smith seems to following the same method of "discovery". He argues that lines divided in this way will be the most beautiful, whether intentionally composed in that way or by accident.
And of course, The Golden Ratio. This one certainly is quite precise, although the degree of adherence to it allowed by different advocates varies. And, the idea that it is important to humans psychologically even though an artist might not realize it consciously is almost ubiquitous (that idea, for example, underlies the argument that the Rule of Thirds gains its power through similarity to this ratio). But the interesting thing, I think, is that modern conception of The Golden Ratio as a rule for aesthetics originated in writings very like Westhoff's. Adolph Zeising, a 19th-century German intellectual, discovered what seemed to be the appearance of the ratio in the branching of plants, and then started finding it everywhere. This touched off a fad of searching for the ratio everywhere, from ancient Greek art and architecture, to the Pyramids, to the Renaissance masters.
Westhoff goes to great pains to claim that this is a "method" of analysis, not a rule for composition, but it really does come down to the same thing.
All of these follow the same basic idea: someone examining works of great art comes up with a mathematical rule, to which special power in aesthetics is ascribed, whether for overall composition or for the most-powerful placement of important details. Samples throughout history are produced, with lines drawn to show amazing correspondence. There's a very strong appeal in having such a rule. Strong composition is very "right brained" and hard to pin down into logical, fixed rules, and it would be so nice if it turned out that there was a simple, mathematical rule that shook it out into left-brained logic after all. Then we wouldn't need to have that uncertain, undefinable thing that is artistic talent; we could just follow the algorithm and great works would invariably result.
There's no such secret, but that doesn't mean that such rules aren't useful. Having forms and limitations are great ways to aid the creation of art. Sonnets have a strong mathematical structure; it doesn't mean that they're mystically the best way to produce a poem, but if you can work within that form, you've got a focus for making good poetry. These rules can all be used in that same way.
Using the diagonal method in the field would be quite difficult without a focusing screen engraved with the proper lines, at least if you are to follow the 0.005%-precision aspect of the rule. One could certainly use it more informally by sighting mentally from the corners of the frame. In fact, Westhoff suggests that many artists do this intuitively already.
The rule explicitly doesn't concern itself with the overall composition, just the placement of details. Westhoff's writing is a bit inconsistent on the point of whether this is useful for aesthetics, or for indication of meaning and emotion without regard for beauty.
Westhoff suggests that it might also be useful in cropping an existing photo. One could use a template with 45° diagonals to ensure that important elements intersect the lines of the rule.
Additionally, if you subscribe to the theory, you could use a template overlaid on existing photographs for analysis; the rule states that the details intersected by these particular diagonal lines have special psychological importance or emotional meaning. So, you could look at your existing works and see if the theory holds true for you, or the works of a favorite artist to see if that artist subconsciously uses the method.