Is it true that $E(h(X,Y,Z)=\iiint h(x,y,z)f(x,y)g(z)\, \mathrm dx\, \mathrm dy\, \mathrm dz$?

Question

Let $X$, $Y$, $Z$ be real random variables with joint density $f(x,y)$ of $X$ and $Y$ and density $g(z)$ of $Z$. Is it true that $$E(h(X,Y,Z))=\iiint h(x,y,z)f(x,y)g(z)\, \mathrm dx\, \mathrm dy\, \mathrm dz$$ or do we need that $(X,Y)$ and $Z$ are independent or something like this?