Let $E$ be a ring spectrum, and $X, Y$ spectra. What can we say about $E_*(X \wedge Y)$ from knowledge of $E_*(X), E_*(Y)$? Ideally I would hope that there would be some sort of Kunneth spectral sequence, for instance there is one in K-theory by a result of Atiyah. It would seem that the necessary condition is being able to embed a space in spaces whose $E_*$-homology is projective or something like that.
(Wikipedia indicates that I should look at Elmendorff-Kriz-Mandell-May, but I wonder if there is something which works for just plain ring spectra.)