A particle can move in any of three directions (that's where the 3 comes from), with kt/2 being the kinetic energy carried by motion on each the x,y or z dimensions. This link gives a short and sweet bit of book work. That Hyperphysics site is good for many things, actually.
There are three translational degrees of freedom, each contributing 1/2kt to the total energy. This from the equipartition theorem.
Taking it back a step the 3kT/2 can be found by equating the two ideal gas equations,one being obtained experimentally(PV=RT) the other being obtained theoretically using kinetic theory(PV=Nmc bar squared/3)