The bandsaw is the usual choice for resawing, but as the Sheriff said, "we ain't got a bandsaw."
So how do you hold a 1" thick piece of wood on edge so the blade can slice it through the middle? You use a nifty shop-built vacuum fence:
Basically it's just a plywood box with holes in one end and a port for the shop vac on the back. It holds on *great*: it takes a solid tug to pull the plank off the surface. Slipping backwards happens more easily, but rather than drilling bigger holes, I just stick a nail in the rightmost hole to give the wood a little lip to ride against.
I didn't have a hole saw the right size for the vac hose, so I put a lid over the port and use velcro to stick the hose to the lid.
Also note how I made the box: rather than fiddling with the edges of the box to make the faces perfectly parallel, I cut some 1" strips for the edges but then used the 3/4" edge for the side rather than the 1" face. Since the plywood has very consistent thickness, the box came out very true.
The holes on the backside were an attempt to make the fence stick to the rip fence, but putting a flat base turned out to be much more stable, and also gave me room for a handle. That makes it easy to hold down and against the rip fence while moving it forward. Remember to take small 1" bites at a time, and to keep the same face against the fence if you have to flip the piece over (when the plank is taller than the height of the blade). It's also good to clamp a piece of scrap above the plank: that prevents the plank from lifting up, and covers most of the extra holes. It also tends to hold the offcut in place after I cut all the way through. I'm not entirely comfortable with that, so maybe I'll start clamping the scrap slightly above the workpiece to give the offcut room to float free.
Dimensions: Vacuum box: 14" x 9" x 3 thicknesses of 3/4" plywood. 1/8" holes spaced at 1" intervals. The 14" length and flat base let me use it with my crosscut sled for even better control.
Oh, it's also very important, after building the box but before installing the base, to put the small holes face down on a flat floor and hook the vac hose to the output port. Hovercraft!