Short answer: turned ionization detectors are easy enough, but they are not "Geiger counters".
The core of a Geiger counter is a gas ionization detector that runs in a saturated cascade mode. They respond to ionization in the gas and are very nearly digital in nature.
Notice that Geiger counters are often rigged to emit an audible click when it respond, and there is no difference between the clicks. That's because the saturated pulse generated by the cascade is the same regardless of the source of amount of radiation.
This energy-insensitive mode of operation has one really big advantage: it's doesn't require careful calibration, which makes is very suited for a basic hand-help survey instrument but completely useless for a particle ID.
You can run the core detector in a linear gain mode, but then it isn't a "Geiger counter", and you need rather more sophisticated electronics backing it. In fact, this is the preferred way to build a sophisticated detector, but it requires careful calibration an the requisite electrons used to be pretty bulky and power-hungry.
There is one thing you can do for resolving different classes of radiation, and that is take advantage of the different penetration depths. At nuclear decay energies (a few 10 of keV up to a few MeV), gammas penetrate more deeply than betas which penetrate more deeply than alphas. By engineering the thickness of the walls you can exclude the alphas or both the alphas and the betas.
However, that is not accurate enough to let you sort middle energy gammas from higher energy gammas (that is to pick out individual lines) because the penetration of any particle ray is subject to considerable random variation.
Most off the shelf Geiger counters have a "thin" window designed to allow betas in. Designing a Geiger style counter to accept alphas is delicate and of limited use because most decay alphas range out completely in a few mm of air.