So I went to Home Depot and bought a couple pieces of craft wood. Which are really cheap and come in (usually) 2 foot or 4 foot lengths and a variety of dimensions. I like to use my miter box and hand saw for smaller projects because it just seems easier that way.
I removed each drawer and just measure the interior spaces and starting working on the grid pattern I wanted. It's really great to have a brad nailer for this job. It makes the assembly a breeze. I suppose you could also use screws, if you are careful and pre-drill. In some places I have used glue only, but over time these joints haven't held up as well.
I just held the pieces in place when shooting the nails in. I wasn't too worried about perfect right angles.
This particular organizer is for my utensils. I haven't created a bottom for it (but you easily could with some thin plywood) because I prefer to put a layer of that puffy shelf liner (that is anti-slip).
You can see the green liner peeking through below.
And I made another one for my utensils. And each drawer organizer only cost (save the brad nailer and air compressor) about $4 in wood.