In my experience, Dirac is most often cited as the origin of this thought experiment. However, from what I've read in his Principles of Quantum Mechanics, he never actually introduces the idea with multiple polarizers, only with a single crystal of tourmaline.
In The Principles of Quantum Mechanics (1930), chapter 1, Dirac introduces a now-standard motivating example for quantum superposition and uncertainty, an experiment using the polarization of light (hypothesized as composed of photons in the development) at oblique angles passing through polarizers (e.g. a crystal of tourmaline). This description, with block quotes from Dirac, seems like a good exposition. The same site has a reproduction of the first chapter. But Dirac's actual development never introduces an oblique polarizer between two other polarizers (one vertical, one horizontal). For instance:
Let us take a definite case. Suppose we have a beam of light passing through a crystal of tourmaline, which has the property of letting through only light plane-polarized perpendicular to its optic axis. Classical electrodynamics tells us what will happen for any given polarization of the incident beam. If this beam is polarized perpendicular to the optic axis, it will all go through the crystal; if parallel to the axis, none of it will go through; while if polarized at an angle $\alpha$ to the axis, a fraction $\sin2\alpha$ will go through. How are we to understand these results on a photon basis?
A beam that is plane-polarized in a certain direction is to be pictured as made up of photons each plane-polarized in that direction. This picture leads to no difficulty in the cases when our incident beam is polarized perpendicular or parallel to the optic axis. We merely have to suppose that each photon polarized perpendicular to the axis passes unhindered and unchanged through the crystal, while each photon polarized parallel to the axis is stopped and absorbed. A difficulty arises, however, in the case of the obliquely polarized incident beam. Each of the incident photons is then obliquely polarized and it is not clear what will happen to such a photon when it reaches the tourmaline.
In our present example the obvious experiment is to use an incident beam consisting of only a single photon and to observe what appears on the back side of the crystal. According to quantum mechanics the result of this experiment will be that sometimes one will find a whole photon, of energy equal to the energy of the incident photon, on the back side and other times one will find nothing. When one finds a whole photon, it will be polarized perpendicular to the optic axis. One will never find only a part of a photon on the back side. If one repeats the experiment a large number of times, one will find the photon on the back side in a fraction $\sin 2\alpha$ of the total number of times. Thus we may say that the photon has a probability $\sin 2\alpha$ of passing through the tourmaline and appearing on the back side polarized perpendicular to the axis and a probability $\cos 2\alpha$ of being absorbed. These values for the probabilities lead to the correct classical results for an incident beam containing a large number of photons.
In fact, it appears as though Dirac's thought experiment serves only as a reconciliation of quantum mechanics with the classical theory of light, rather than as a motivating example of quantum mechanics, as it is presented e.g. in the website linked, among other references (if this claim is dubious I can provide examples). The three polarizer experiment, on the other hand, does motivate a quantum mechanical description, and hence is very different from Dirac's, though it can be deduced without much trouble.
When was this polarized photon description first stated in the familiar "three polarizers" format? Or, what are the earliest examples that were influential in quantum mechanics pedagogy?