If you form the smash product X = K(A,p) n K(B,q) of two Eilenberg-MacLane spaces (I am using "n" for the smash product symbol here), then the resulting space is (p+q-1)-connected, and the first non-trivial homotopy group in dimension p+q is A \otimes B.

To see this "geometrically", I would model the EM spaces as CW-complexes, whose first non-basepoint cells are in dimensions p and q respectively.  Then in the smash product X, the bottom dimensional cells are products of the bottom dimensional cells of the EM spaces; this gives the connectivity result, and by looking at the attaching maps of the (p+q+1)-dimensional cells in X, you can compute \pi_{p+q}.  Now you can use obstruction theory to produce a map X --> K(A\otimes B, p+q).