As discussed previously, a Geiger counter is a fairly simple circuit that takes a high voltage and runs it through the switch-like Geiger-Müller tube and into the meter mechanism. To build a Geiger counter, we need to look at three basic parts: a high voltage source, the GM tube itself, and the counting mechanism.
DIY Geiger Counter Circuit
After a bit of research, I decided to base my own circuit on this design by markusb on RobotShop.com. In this circuit, built around a Röhre ZP-1320 GM tube, the high voltage source is provided by a 40:1 transformer and charge pump that supplies the requisite 500V. Of course, a transformer requires an AC input, so a 555 timer in astable mode feeds an alternating 5V to the transformer. The metering side of the circuit uses a 555 timer to generate an electrical pulse that can be fed into a microcontroller or analog counter.
For Project Pripyat, I'm using a Soviet-era SBM-20 (СБМ-20) GM tube that we had lying around the shop, and I'll need to adjust the circuit somewhat to power it. I like the oscillator-transformer concept, and I'll keep that in tact, but I think I can simplify the charge pump somewhat and still provide a reliable 400V to power my tube.
I also want to run my Geiger counter off a rechargeable LiPo battery, so I'll add a 3.7V pack and an Adafruit Powerboost 500 module to provide a stable 5V (and handle battery charging). On the counter end, I'd like to be able to have extended functionality such as data logging or triggering various outputs, so I'm going to send the GM tube pulse to an Arduino Nano (after passing through a voltage divider, of course).
After quite a bit of trial and error, I've come up with this: 
As you can see from the schematic, I'm using a different transformer and a simpler rectifier circuit than the model's diode ladder. I also tweaked the oscillator slightly, using only an N-channel MOSFET instead of the NPN-MOSFET combination in the original design. The NPN transistor, though, serves a new purpose as the pulse generator that drives the digital input for the Arduino. From this configuration, I can add a piezo buzzer, LED, analog meter, or any other output as well as save data to memory or pipe it to a computer via serial connection.
There are a couple of important things to note with this circuit: First, I have to reiterate that it is a high voltage circuit and you will likely get popped pretty hard if you're not paying attention. I accidentally touched one of the capacitor terminals on the charge pump during testing and received quite an unpleasant surprise! It's unlikely that you will suffer any lasting damage, though, but caution is the order of the day whenever high potentials are involved. Second, if you choose to use a different GM tube for this circuit, you will need to adjust the resistor and capacitor values in the charge pump. The film capacitors that I used are rated for up to 700V and the first version I assembled (using 5 capacitors) built a potential in excess of 600V (and literally screamed at me). Third, and it should go without saying, this is a device used to measure ionizing radiation and ionizing radiation is a hazardous phenomenon. Please take all precautions to limit your exposure to beta and gamma particles by using alpha sources for testing and storing your radioactive samples in appropriately shielded containers.