  We introduce the notion of 321-avoiding permutations in the affine Weyl group
$W$ of type $A_{n-1}$ by considering the group as a George group (in the sense
of Eriksson and Eriksson). This enables us to generalize a result of Billey,
Jockusch and Stanley to show that the 321-avoiding permutations in $W$ coincide
with the set of fully commutative elements; in other words, any two reduced
expressions for a 321-avoiding element of $W$ (considered as a Coxeter group)
may be obtained from each other by repeated applications of short braid
relations.
  Using Shi's characterization of the Kazhdan--Lusztig cells in the group $W$,
we use our main result to show that the fully commutative elements of $W$ form
a union of Kazhdan--Lusztig cells. This phenomenon has been studied by the
author and J. Losonczy for finite Coxeter groups, and is interesting partly
because it allows certain structure constants for the Kazhdan--Lusztig basis of
the associated Hecke algebra to be computed combinatorially.
  We also show how some of our results can be generalized to a larger group of
permutations, the extended affine Weyl group associated to $GL_n({\Bbb C})$.
