This is not a very impressive discovery… but I figured out that with a (very little) bit of work, a PATA ribbon cable works perfectly with the Raspberry Pi B+!
However, as you can see here, PATA cables have one pin sealed off, which blocks it from plugging in to the GPIO pins on the Pi.
As it turns out, it is just a thin piece of plastic, and has a usable connection underneath.
I used the nice ghetto method of hammering a nail into it.
And there you have it – a nice ribbon cable that fits perfectly into the Pi GPIO pins.
Also note – If you have a case like mine, you might have to break off the top of the PATA connector for the lid to fit properly.