When can a finite map be blown up to a flat one? Let $f:X\to Y$ be a generically finite proper morphism of varieties.  There is some locus in $Y$ over which the fiber of $f$ is positive dimensional, so we blow it up, along with the preimage of it in $X$ to get a map $\tilde{f}:\tilde{X}\to\tilde{Y}$ which has finite fibers.
Are there any nice conditions that will guarantee that the map $\tilde{f}$ is flat?
 A: Since the blowups are proper and $f$ is proper, the "property P argument" shows that $\tilde f$ is proper. A proper quasi-finite morphism is finite (EGA IV 18.12.4), so $\tilde f$ is finite.
This (more or less) reduces to the case when $f$ is finite to begin with, so no blowups are needed. Then the only condition I know to ensure flatness is the one Dave Anderson cited, [Matsumura's Commutative Ring Theory, Theorem 23.1]: if $Y$ is regular and $X$ is Cohen-Macaulay, then $f$ is flat.
A: Have you seen "Critères de platitude et de projectivité. Techniques de "platification'' d'un module" by Raynaud-Gruson (1971)?  In particular 5.2.2.  I think it is very close to what you want (this was explained to me by Bhargav Bhatt not too long ago).  It doesn't say that any blow-up works, but there is one that's ok.
Basically, there exists a blow-up $Y' \to Y$ (you can assume $Y'$ is normal, ie normalize the blow-up) such that the appropriate component(s) of the fiber product $Y'' \to Y' \times_Y X$ give us a map $Y'' \to Y'$ which is flat.  This is proven in the modern setting by Hilbert-Scheme arguments usually (also see for example various papers talking about universal flattening).
