When does a cohomology theory have a ring structure?

I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW structure and I simply couldn't find any reasonable definition for the cup product on CW spaces. Does such a thing exist? If not is there some conditions on a cohomology theory that allows for a natural definition of a cup product? Obviously you could define some contrived cup product structure via singular or simplicial homology, but what about Cech Cohomology or any of the extraordinary cohomology theories?

The ring structure in cohomology arrises in the following way. We have a natural map $$ \triangle:X\rightarrow X\times X $$ which sends $x\mapsto (x,x)$ the diagonal. Thus in cohomology, we get a nice map $$ \triangle^*:H^k(X\times X)\rightarrow H^k(X), $$ since cohomology is contravariant. Now, for nice cohomology theories one can make sense of a map $$ P=H^i(X)\times H^j(Y)\rightarrow H^{i+j}(X\times Y) $$ Taking $X=Y$ one can compose these maps to get a ring structure $$ \triangle P:H^i(X)\times H^j(X)\rightarrow H^{i+j}(X). $$ This is explained a bit in Hatchers book http://www.math.cornell.edu/~hatcher/AT/AT.pdf on page 185. So for any cohomology theory you have to be able to make sense of the map $P$.