  Using chiral perturbation theory we investigate the QCD shear viscosity to
entropy density ratio below the deconfinement temperature (~170 MeV) with zero
baryon number density. It is found that the viscosity to entropy density ratio
of QCD is monotonically decreasing in temperature (T) and reaches 0.6 with
estimated ~50% uncertainty at T=120 MeV. A naive extrapolation of the leading
order result shows that the ratio reaches the 1/(4 pi) minimum bound proposed
by Kovtun, Son, and Starinets using string theory methods at T~200 MeV. This
suggests a phase transition or cross over might occur at T less than 200 MeV in
order for the bound to remain valid. Also, it is natural for the ratio to stay
close to the minimum bound around the phase transition temperature as was
recently found in heavy ion collisions.
