I was just going to cut the walnut at 45's and glue and brad nail it all together. Unless you have other suggestions, Mr. Wood.
That's exactly what I would have done, except the brad nail. Nails do hold things together, but over time, they loosen due to expansion and contraction of the wood. Plus that's one more thing to sand and fill with wood puddy. I would use splines to strengthen the joint, but without a spline cutting jig, it would be pretty hard to do.
Where the sides meet the bottom is a butt joint, not super strong without some other kind of re-enforcement. Maybe try using dowels or biscuit to make it a bit stronger. If I were making the box, I would float the bottom using dados in the sides, or rabbet the bottom to add more surface area for the glue.