Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.  

**Question:** Is $Y$ also Cohen-Macaulay?  Are there common conditions which imply it is?  

If $Y$ is not normal I know of several ways to show that the answer is no.  
There are obvious spectral sequences but I don't see how to deduce what I want from them, perhaps I'm being dumb (or maybe there is an obvious example).