There is more than one possible generalization. The most common is the *universal coefficient spectral sequence*. Given a (homotopy) commutative ring spectrum $E$ and a spectrum $X$, there is under certain conditions a spectral sequence
$$ Ext^{p,q}_{E_*}(E_*(X), E_*) \Rightarrow E^{q-p}(X) $$
This is true for example if

- $E$ is an $A_\infty$-ring spectrum (e.g. if $E =ko, ku, KO, KU, TMF, MO, MSO, MSpin \dots$) [EKMM, IV.4]
- $E$ is even and Landweber exact (e.g. if $E = MU, E(n), E_n, \dots$) [Adams' lectures on generalized cohomology, which Peter alluded to, and Devinatz: Morava Modules and Brown-Comenetz Duality, Prop 1.3, and the discussion thereafter]

Sometimes, this spectral sequence is not extremely useful though. For example, take $E = KO$. In general, $KO_*X$ might have infinite cohomological dimension over $KO_*$ so that the spectral sequence might be difficult to control. In this case another perspective is more useful: Anderson duality.

Consider the functor $E \mapsto Hom(\pi_*E, \mathbb{Q}/\mathbb{Z})$ from the homotopy category of spectra to graded abelian groups. As $\mathbb{Q}/\mathbb{Z}$ is injective, this is a cohomological functor, by Brown representability represented by a spectrum $I_{\mathbb{Q}/\mathbb{Z}}$. There is an evident map $H\mathbb{Q} \to I_{\mathbb{Q}/\mathbb{Z}}$ whose fiber we denote by $I$. For a spectrum $E$, we define its *Anderson dual* $IE$. to be the function spectrum $F(E, I)$. It is easy to show that we get for every spectrum $X$ a short exact sequence
$$ 0 \to Ext^1_{\mathbb{Z}}(E_{k-1}X, \mathbb{Z}) \to (IE)^kX \to Hom_{\mathbb{Z}}(E_kX, \mathbb{Z}) \to 0.$$
That means that there exists always a short exact sequence computing from $E$-homology the $IE$-cohomology. This is, of course, only useful if we can identify $IE$. Luckily, this has been done for a few spectra:

- $IH\mathbb{Z} \simeq H\mathbb{Z}$
- $IKU \simeq KU$
- $IKO \simeq \Sigma^4 KO$ [see e.g. Heard, Stojanoska]
- $ITmf \simeq \Sigma^{21}Tmf$ (at least at primes $>2$) [see Stojanoska]

For example, for $KO$ this means that we get a universal coefficient sequence
$$ 0 \to Ext^1_{\mathbb{Z}}(KO_{k-1}X, \mathbb{Z}) \to KO^{k+4}X \to Hom_{\mathbb{Z}}(KO_kX, \mathbb{Z}) \to 0.$$