To find the eccentricity of the hyperbola given by the equation $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$, we break down the solution into detailed steps:

1. Identify the formula of the hyperbola: Given $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$, we can see that it is in the standard form of a hyperbola centered at the origin with the $x$-axis as its transverse axis.

2. Determine the values of $a$ and $b$: From the equation, we can extract that $a^2=4$ and $b^2=2$, which leads to $a=2$ (since $a$ is the semi-major axis and is always positive) and $b=\sqrt{2}$.

3. Calculate the focal length $c$: For a hyperbola, the relationship between $a$, $b$, and $c$ (the distance from the center to a focus) is given by $c^2=a^2+b^2$. Substituting the values of $a$ and $b$, we get:
$$c^2 = 2^2 + (\sqrt{2})^2 = 4 + 2 = 6$$
Thus, $c = \sqrt{6}$.

4. Find the eccentricity $e$: The eccentricity of a hyperbola is given by the formula $e=\frac{c}{a}$. Substituting the values of $c$ and $a$, we get:
$$e = \frac{\sqrt{6}}{2}$$

Therefore, the eccentricity of the given hyperbola is $\boxed{\frac{\sqrt{6}}{2}}$.