TITLE: Hn(X, OX) = 0 for X birational to a regular affine variety?
QUESTION [7 upvotes]: It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.
If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}_Y) = H^n(X, \mathcal{O}_X)$. (right?)
I have not seen any examples of blow-ups of smooth varieties (with arbitrary center) whose fibres are not unions of projective spaces (sub-question: do such blow-ups exist?).
So I am lead to wonder: is it always true that $H^n(Y, \mathcal{O}_Y) = 0$ if $f: Y \to X$ is birational, $Y$ is integral, and $X$ is smooth and affine?
Edit: I mean't to include the hypothesis that $f$ is proper, and that $n$ is any integer $> 0$.

REPLY [4 votes]: As pointed out above in the comments by Piotr Achinger, I'm going to assume you meant the map to be proper.  
For the first question then:  sure, the fact that $R^i f_* O_Y = 0$ for $i > 0$ implies your first assertion that $H^n(Y, O_Y) = H^n(X, O_X)$ (here $f$ denotes the map $Y \to X$, you assumed $X$ was affine, but you don't need it for this).  This condition is basically that the non-singular $X$ has rational singularities.  In characteristic zero, this is a fairly straightforward application of Grauert-Riemenschnedier vanishing and holds as long as both $Y$ and $X$ are smooth (it doesn't matter whether you did blowups at smooth centers or not).  Interestingly, we still don't know whether this vanishing holds for regular $Y$ and $X$ in mixed characteristic (although we do know this vanishing for regular $Y$ and $X$ in positive characteristic http://front.math.ucdavis.edu/0911.3599.
For the second question "fibres are not unions of projective spaces", you've already seen one non-normal example.  Ulrich gives a nice example of a non-rational component in the comment below (originally I said something stupid here).
The last question, "is it always true that $H^n(Y,O_Y)=0$?", here's an example which this time is normal.  It is Section III of Cutkosky's A new characterization of rational surface singularities
Note the $R^2 h_* O_Z \neq 0$ in the papers notation is exactly the non-vanishing you were interested in.
Note:  In the interest of full disclosure, this paper was pointed out to me by Hailong Dao in this answer on math overflow.