Why do we have different signs before the delta on the Klein-Gordon and the Dirac Green's function equation? Let's read equation (2.56) on Peskin & Schroeder 
$$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y).$$
Let's look now to equation (3.118)
$$(i\gamma^{\nu}\partial_{\nu}-m)S_R(x-y)=i\delta^4(x-y).$$
Why the different sign before the delta? and when I consider the propagator for any other field theory, is there a way to know what sign is to be used?
 A: Maybe the answer is connected with the fact that propagator is the inverse of Lagrangian operator.
An action of free theory may be written as
$$
S[\psi ] = \int d^{4}x (\psi^{a})^{*}\Delta_{ab}\psi^{b}.
$$ 
Here $()^{*}$ means conjugation which leaves $(\psi^{a})^{*}\Delta_{ab}\psi^{b}$-form lorentz-invariant. For example,
$$
S_{KG}[\varphi ] = \frac{1}{2}\int d^{4}x \left( (\partial_{\mu}\varphi )^{2} - m^{2}\varphi^{2}\right) = \frac{1}{2}\int d^{4}x \varphi (-\partial^{2} - m^{2}) \varphi , 
$$
$$
S_{D}[\bar{\psi}, \psi ] = \int d^{4}x \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi ,
$$
$$
S_{EM}[A] = \int d^{4}x \left( -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} - \frac{1}{2\alpha}(\partial_{\mu}A^{\mu})^{2}\right) = 
$$
$$
=\frac{1}{2}\int d^{4}x A^{\mu} \left( \partial^{2}g_{\mu \nu} - \left[ \frac{1}{\alpha} - 1\right]\partial_{\mu}\partial_{\nu}\right)A^{\nu}.
$$
The propagator is the inverse lagrangian operator:
$$
D_{ab}(p) = \frac{1}{\Delta_{ab}(p)}.
$$
So now you know how to get the sign in equation for propagator. 
Maybe you want to know how to get the sign in lagrangian. The sign (in general - scaling factor) of lagrangian operator doesn't affect on the equations of motion (as well as the polarization sum rule $\sum_{s}u_{a}^{s}(u_{b}^{s})^{*}$ is determined from equations of motion only up to the scaling factor). But it is important because it determines the sign of kinetic part in lagrangian ($\bar{\psi}\gamma^{0}\partial_{0}\psi$ for Dirac case, $\partial_{0}A^{i}\partial_{0}A^{i}$ in EM case, $(\partial_{0}\varphi)^{2}$ in scalar case etc.). This affects on unitarity of theory, so, of course, is very important. 
