Given a complex oriented cohomology theory $E$, one can define a formal scheme of Weil pairings on the associated formal group, as explained in the paper "Weil pairings and Morava $K$-theory" by Matthew Ando and me. If we let $R_E$ denote the ring of functions on this scheme, then there is a natural map $R_E\to E^*K(\mathbb{Z},3)$. This is an isomorphism if $E$ is Morava $K$-theory or Morava $E$-theory. I think that it is also an isomorphism for $E=MU$ or $E=kU$ but not for $E=H$. However, there is a natural short exact sequence
$$ kU^*K(\mathbb{Z},3)/v \to H^*K(\mathbb{Z},3) \to \text{ann}(v,kU^*K(\mathbb{Z},3)) $$
(where $v$ is the standard generator of $\pi_2kU=kU^{-2}(\text{point})$). I think that this is probably an effective way to understand $H^*K(\mathbb{Z},3)$.

Some other things that are going on in the background here:

- There is a fibration $K(\mathbb{Q}/\mathbb{Z},2)\to K(\mathbb{Z},3)\to K(\mathbb{Q},3)$. Here $K(\mathbb{Q},3)$ is the rationalisation of $S^3$ and is not so hard to understand.
- $K(\mathbb{Q}/\mathbb{Z},2)$ is the colimit of the spaces $K(\mathbb{Z}/n,2)$.
- One can understand $K(\mathbb{Z}/n,2)$ using the multiplication map $K(\mathbb{Z}/n,1)\times K(\mathbb{Z}/n,1)\to K(\mathbb{Z}/n,2)$.