This is how to make a platform bed to go along with this Instructable's woven headboard. Building it was actually way easier than I was anticipating, and after a somewhat fretful first night (I'd never built a bed before, and was half-expecting it to collapse on me), is pretty sturdy. It's relatively inexpensive to make: the supplies cost around $120, and can be finished in a weekend.

Step 1: Supplies!

2 pieces, 1/2" x 4' x 8' hardwood plywood
2 pieces, 7/16" x 4' x 8' OSB sheathing
7 pieces, 1 x 3 x 8" lumber
3 pieces, 2 x 4 x 10" lumber
3 rolls of 1" wood veneer
1 box, 2" #8 screws
Wood glue
Clamps
Sandpaper
Wood stain (optional)
Finish
1" furniture sliders

Step 2: Cuttin' Wood!

Hardwood Plywood
Cut the sheets into:
2 pieces, 15 x 81"
2 pieces, 15 x 61"
4 pieces, 15 x 20"
4 pieces, 15 x 25"

OSB Sheathing
Cut the sheets into:
2 pieces, 30 x 80"

1x3s
Trim to 5' length

2x4s
Cut to:
2 pieces, 4.5' length
2 pieces, 6' length

Step 3: Platform!

Once you've got the OSB cut to the right size, you're pretty much done with it until it comes time to put everything together. Unless you want to make it fancy, that is.

I spray-painted the top of each piece black, to kinda make it less obvious when in use (there's about a 1/2" gap between the mattress and the frame when it's all assembled). Doesn't need to be pretty, just not looking like its cheap pressboard self.

I then took 2 pieces of leather, about 1 x 4", folded that in half, superglued it to itself, then superglued it to the top of each platform half to make a little handle to help get it up off the support beams. I also put a couple small nails through each strap to give it a little better grip.

Step 4: Bed Exterior!

Veneer!
Apply veneer to 3 sides of each hardwood piece (no need to do the bottom, no one'll see it). Cut it to length, leaving a smidge extra on each side (you'll need to do it in stages, 2 sides, then the remaining side). Use an iron to slowly press it on; do not rush this. You want that glue real hot, and you want the wood strip pressed down real well. Then let it sit for a while.

Once it's had a chance to cool and set, use a utility knife to trim the excess.

I did not do this step in this order. Learn from my mistakes.

Glue!
Glue and then clamp a 15 x 20" piece to either end of each 15 x 61" piece; glue and clamp a 15 x 25" piece to either end of each 15 x 81" piece. Make sure the edges are as flush as possible, especially the bottom. Let these dry for at least a couple hours.

Step 5: Support Rails!

Sides
Measure your 1x3s to get the exact thickness; measure your OSB sheathing to get the exact thickness; add those 2 thicknesses together--that's how far down from the top edge of the side you want the top of the 2x4 rail to hit.

Clamp a 6' 2x4 at that distance from the top edge, and get it level. Drill pilot holes (make sure not to drill through the front; be especially careful in the center where there's not the extra thickness).

Unclamp the 2x4, apply glue, and reclamp it; again, make sure it's level. Screw it into the side piece. You don't need to worry about screwing the front on the ends, but there's about 2 millimeters breathing room in the middle, so exercise caution.

Head/Foot
For the head and foot, you just need the thickness of the OSB. Clamp a 5' 2x4 at that distance from the top edge, then attach it the same way as with the sides.

Step 6: Test Assembly!

Put the sides together (butt joints, foot/head between the sides) and add the 1x3 supports. Assuming everything fits as planned, drill pilot holes in each end of the sides, 1 at top, 1 at bottom. I also decided to screw 3 of the 1x3s to the frame (head, foot, and middle) to give a little more structural integrity, so drilled pilot holes for that as well.

Step 7: Sand, Stain, Finish!

Sand the exterior-pieces with high-grit paper. Be careful not to go through the veneer on the edges or on the plywood.

If you're going to stain, do it now.

Then apply your finish as the can directs.

Step 8: Assemble!

Install the furniture sliders on the bottom of your frame pieces, about 2" from each end.Get the screws for them as centered as you can.

Take all your bits to your bedroom and put 'er together! If you made the headboard that goes with this (or any headboard, for that matter), mount that first. If you have a cat to help you with this part by jumping on things, all the better.

Now, take a nap!