In dimension 2, a rational map becomes a morphism after a sequence of blowups. Does this still hold in higher dimensions?

Yes, but this is a very difficult result, due to Hironaka. To be precise : given a rational map $f:X > Y$, there exists a birational morphism $b:\hat{X}\rightarrow X$, obtained as the composition of successive blownup with smooth centers, such that $f\circ b$ is a morphism. 

