Here's one characterization that I learned from Serre
(see Definition 7.1.1 in his
<a href="http://www.math.mcgill.ca/~darmon/pub/Articles/Serre/c.pdf">*Topics
in Galois Theory*</a> (p.65)): an element $g$ of a finite group $G$
satisfies $\chi(g) \in {\bf Q}$ for all characters $\chi$ **iff**
$g$ is conjugate in $G$ to $g^m$ for all $m$ relatively prime to 
the exponent $e(g)$.  [If $m$ is not coprime to $e(g)$ then `$e(g^m)<e(g)$`
so $g^m$ cannot possibly be conjugate to $g$.]
It is enough to check this for all $m$ relatively prime to $\left| G \right|$.  In particular, all character values are rational **iff** every group element
is conjugate to its $m$-th power for all $m$ coprime to $\left| G \right|$.