For starters I decided to go with a 5' x 5' box (25 squares), so I drove to our local Home Depot ("more saving, more doing" and whatnot) to buy the materials. I am somewhat worried about the 5' x 5' box being too big to access the middle plots without stepping on the soil (one of Mel's big no-nos). I don't think this will be the case, but I may explore leaving the middle plot empty as a place to stand while tending and harvesting my bounty...
Anyways, my materials list was pretty short: boards, screws, a ground cloth, and some rope. For the boards, I bought 2 10" x 2" x 10' boards (non-pressure treated- you don't want any of the chemicals leaching into the soil) and had them each cut in half at the store, giving me the 4 sides of the beds. 10" deep was perfect- 6" is recommended by Mel, but because I'm trying to grow some root vegetables (more on them later) I wanted more depth. I also didn't want the soil to be right up to the top of the box, as it would be with a 6" board.
Deck screws are recommended by Mel, but instead I settled on regular screws because the deck screws had an ominous warning on the box about not using them within 5 miles of any body of water, or something like that. I used 12 in total, drove them in with a cordless drill, and all the sides looked like this:

I also dug around in the toolbox and found 8 "L" braces (I think they were left over from some IKEA contraption. Maybe our wobbly dresser?) which I used to reinforce the insides of the boxes:

This gives me a pretty sturdy box which I will be able to (potentially) move around in future seasons or depending on the area with best light (more on this later). I also measured off 1 foot increments all around the top of the frame and put in screws so I can tie off ropes to get the squares. Mel says that you should use a wood lattice because ropes aren't sturdy or durable enough, and he's probably right. I think, however, having ropes allows me to change up the grid for future plantings. I used a heavy duty braided nylon nautical line (to withstand rot) which I think will stand up, at least for a season.
PS- It was a good thing I made the box so sturdy. I came home the other day, puttered around for while, then realized the box was gone. I was utterly dumbfounded because I just couldn't figure out who, or why, would someone steal a 5' x 5' frame? Then I looked over and saw it lying up against the wall of our neighbor's garage. WTF? Turns out, one of our neighbor's daughters had come over the other day (I don't think her parents can speak English) and asked if I was going to use the "pile of wood" in the backyard and, if not, could she have it? I assumed she meant the bunch of fallen sticks and branches I had gathered from our yard, so I said sure. That wasn't what she meant. She apologized, and I dragged the box back around the fence and its none the worse for wear. They're probably going to spend the rest of the summer wondering why el gringo is trying to grow food inside a perfectly good bed frame...