How can I support the bottom of a long, open cabinet?
I like the open look but to tell the truth you will need some (probably 2 or more brackets) this method, or some decorative dowels that can lift the load. 

I have done similar and ended up drilling the center of the dowel and using 1/4” all thread with washers and nuts above and below. to hide the all thread at the bottom I found some brass loops to screw on the bottom end. (Think ceiling toggle bolt but very high end).

The dowels still make the space appear open you could even use a small chunk of the same wood the box is made of 3/4” square x 12” and drill that out and use all thread. 

I did try screwing directly into the dowel but with a load (cook books) the screws started pulling out. After adding the all thread it was strong but I needed to hide the nut (I only had counter sank the holes enough for the screw heads and did not want to weaken the area further so the nut and a little of the all thread were exposed, the loops worked out great, if you wanted to reduce clutter a small ball or other type of device could be screwed on to hide the nut and threads.