I would like the look of it better with the grain on the sides running up and down and then, you would need to use dowel joints or some other joint that would yield a strong glue joint. I suspect the reason this project was was built this way was to make it simple and to mere allow gluing the pieces together because you are gluing face grain to edge grain. With grain on the sides running vertically you are gluing face grain to end grain and the result wouldn't be very strong.

If you want to use dowels and keep this simple, look for a set of dowel centers. Woodcraft sells a set of 8 centers; 2 @ 1/4", 2 @ 5/16", 2 @ 3/8", and 2 @ 1/2". Drill holes in the vertical piece (end grain in this case); one on end, slip in the dowel centers, line up the top piece with the edges of the side, and push down on the dowel centers. The points will leave an indent where the center of the dowel hole should be drilled in the mating piece. I'd practice on scrap first, but it's a simple process. Before I got complex and acquired way too many tools, I used dowels and the centers worked great. Just a suggestion to keep it simple.