  We establish analogues of the Hermite-Poulain theorem for linear finite
difference operators with constant coefficients defined on sets of polynomials
with roots on a straight line, in a strip, or in a half-plane. We also consider
the central finite difference operator of the form $$ \Delta_{\theta,
h}(f)(z)=e^{i\theta}f(z+ih)-e^{-i\theta}f(z-ih), \quad\theta\in[0,\pi),\ \
h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire
function of a certain kind, and prove that the roots of $\Delta_{\theta, h}(f)$
are simple under some conditions. Moreover, we prove that the operator
$\Delta_{\theta, h}$ does not decrease the mesh on the set of polynomials with
roots on a line and find the minimal mesh. The asymptotics of the roots of
$\Delta_{\theta, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial
$p$. Some other interesting roots preserving properties of the operator
$\Delta_{\theta, h}$ are also studied, and a few examples are presented.
