It's very easy to build a geiger counter from various parts.
A Geiger counter consists of a power supply, a tube containing a gas that ionizes when a particle passes through it and a speakeror even a galvanometer.
The first thing to do is to buy a Geiger tube. Do not try to make one.
The main difficulty is that we must supply it with "high voltage".
To resolve this problem, I bought EL-Wire.
http://en.wikipedia.org/wiki/Electroluminescent_wire
Electroluminescent wire (often abbreviated to EL wire) is a thin copper wire coated in a phosphor which glows when an alternating current is applied to it.