The Künneth formula for ordinary homology as you present it works only when $R$ is a PID (or more generally of cohomological dimension 1).
For a general well-behaved homology theory[1] (this includes both ordinary cohomology, K-theory and cobordism) there is a Künneth spectral sequence
$$E^2_{p,q}=\mathrm{Tor}_{p,q}^{E_*}(E_*X,E_*Y)\Rightarrow E_{p+q}(X\times Y)$$
(see for example theorem 4.1 in
Elmendorf, A. D.; Kříž, Igor; Mandell, Michael A.; May, J. P., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Monographs. 47. Providence, RI: American Mathematical Society (AMS). xi, 249 p. (1997). ZBL0894.55001.
with $M=E\wedge \Sigma^\infty_+X$ and $N=E\wedge \Sigma^\infty_+Y$).
When $X$ and $Y$ have the homotopy type of finite CW-complexes, you can use Spanier-Whitehead duality to obtain the same result for cohomology $E^*X=E_{-*}(\mathbb{D}X)$, with a bit of care due to the signs that appear from the duality. So under these hypotheses we obtain a spectral sequence
$$E_2^{p,q}=\mathrm{Tor}^{E_*}_{-p,-q}(E^{-*}X,E^{-*}Y)\Rightarrow E^{p+q}(X\times Y)\,.$$
The case of the Künneth formula corresponds to the degenerate situation in which $\mathrm{Tor}^{E_*}_i=0$ for $i\neq 0,1$.
[1] Precisely, I need the homology theory to be represented by an $E_1$-ring spectrum.