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).

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 to the first question 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).