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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.)

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I'll add more to this later, but:

As far as the classical Kunneth formula, this is a very special thing to ask for. I think that essentially the only spectra that satisfy such a thing are like the Morava $K$-theories and the Eilenberg-Maclane spectra over fields (or PIDs...). (So for example, complex $K$-theory is special because it's determined by all the $K(1)$ theories.)

For a spectral sequence I'm not sure off the top of my head, but I'll get back to you later tonight when I have a moment!

EDIT: Actually EKMM do a very good job of describing the history of such results on page 32 of http://www.math.uchicago.edu/~may/PAPERS/Newfirst.pdf , as you suspected.

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    Really you can just say "Morava K-theories $K(n)$" (for $n \in [0,\infty]$), which is slightly more satisfying.2012-04-29