If $K$ has characteristic $p>0$, an
[Artin-Schreier extension][1]
$L/K$ is cyclic of degree $p$, and not of the indicated form. Such
an $L$  has the form $K(a)$ for a root $a$ of a polynomial of the 
form $X^p - X - \alpha$ for a suitable $\alpha \in K$; the polynomial
then splits over $L$ since its roots are all $a+i$ for $i \in \mathbf{F}_p$
(in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.


**Edit:**
whoops. Looking at Georges Elencwajg answer, I see that I made what he labels as the "First Interpretation" of the question (which was actually a bit sloppy...).


If $K=\mathbf{F}_p(t)$ for transcendental $t$, and if $L/K$ is the Artin-Schreier extension obtained by adjoining to $K$ a root a of $X^p - X - t$, then for $i\ge 1$, $a^{p^i} =a+f(t)$ where $f(t) \in \mathbf{F}_p[t]$  is a polynomial of degree $p^{i-1}$. This show that $a^N \not \in K$  for any $N\ge 1$. So this cyclic extension gives an example meeting Elencwajg's "Second Interpretation" of the question.


  [1]: http://en.wikipedia.org/wiki/Artin%25E2%2580%2593Schreier_theory