There are several ways of looking at explicit representatives of $H^3(G,A)$, and a lot depends on your purpose.
A standard $3$-cocycle is a function on $G \times G \times G$ with values in $A$, so this can be difficult to work with.
Otherwise we look for a small free resolution of $G$, and then use tensor products of chain complexes to get a free resolution of the product. A small resolution of $Z_2$ has in dimension $n$ the group ring $Z[Z_2]$ and boundaries alternatively multiplication by $1+t, 1-t$ where $t$ is the generator of $Z_2$.
With coefficients in $S^1$, which is an injective abelian group, the Universal Coefficients Theorem gives, since Ext$(-,S^1)=0$, that
$$H^3(G,S^1) \cong Hom (H_3(G), S^1), $$
which could be useful.
There are other representatives of elements of $H^3(G,A)$, namely as equivalence classes of so-called "crossed sequences"
$$0 \to A \to M \to^\mu P \to G \to 1$$
where $\mu: M \to P$ is a crossed module. It would be nice to think that this could be useful in physics! However this result is used mainly when $A$ is $G$-module, and in particular a discrete abelian group; so I have not seen such constructions when $A=S^1$.
You may find the work of Graham Ellis on Homological Algebra Programming relevant: