It is known that Fourier integral operators arising when solving
Schr\"odinger-type operators are bounded on the modulation spaces $\cM^{p,q}$,
for $1\leq p=q\leq\infty$, provided their symbols belong to the Sj\"ostrand
class $M^{\infty,1}$. However, they generally fail to be bounded on $\cM^{p,q}$
for $p\not=q$. In this paper we study several additional conditions, to be
imposed on the phase or on the symbol, which guarantee the boundedness on
$\cM^{p,q}$ for $p\not=q$, and between $\cM^{p,q}\to\cM^{q,p}$, $1\leq q<
p\leq\infty$. We also study similar problems for operators acting on Wiener
amalgam spaces, recapturing, in particular, some recent results for metaplectic
operators. Our arguments make heavily use of the uncertainty principle.