Let me work over $\mathbb{C}$ for simplicity. We have $L(V) \cong V^{\otimes n} \otimes_{S_n} \text{Lie}(n)$ where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad; this is, to be clear, true for any $V$, in great generality, and is a special case of a very general fact about free algebras over operads $O$, namely that they are given by
$$V \mapsto \bigoplus_{n \ge 0} V^{\otimes n} \otimes_{S_n} O(n).$$
$\text{Lie}(n)$ is known to be isomorphic as an $S_n$-representation to the induced representation $\text{Ind}_{C_n}^{S_n} \chi$ where $\chi$ is the $1$-dimensional representation $k \mapsto e^{ \frac{ 2 \pi i k}{n} }$ (I don't have a reference for this or know who it's due to). This gives
$$\begin{eqnarray*} L(V) &\cong& V^{\otimes n} \otimes_{S_n} \text{Ind}_{C^n}^{S_n} \chi \\ &\cong& V^{\otimes n} \otimes_{C_n} \chi \end{eqnarray*}$$
which is equivalent to Klyachko's theorem when $V$ is purely even. The statement is just the same if $V$ has an odd component except that the action of $C_n$ on $V^{\otimes n}$ has some different Koszul signs to it. Explicitly, if $v_1 \otimes \dots \otimes v_n \in V^{\otimes n}$ and each $v_i$ is homogeneous of degree $|v_i|$ then the cyclic permutation which commutes $v_1$ past all the other vectors produces
$$(-1)^{|v_1|(|v_2| + \dots + |v_n|} v_2 \otimes \dots \otimes v_n \otimes v_1.$$
For example if the $v_i$ are all odd then the sign is $(-1)^n$ which matches the sign of the $n$-cycle as a permutation in $S_n$.
I guess we can be more specific as follows. If $V$ is purely even then Schur-Weyl duality gives
$$V^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S_{\lambda}(V) \otimes M^{\lambda}$$
where $S_{\lambda}$ is a Schur functor and $M^{\lambda}$ is a Specht module. This gives that $V^{\otimes n} \otimes_{C_n} \chi$ is a sum of Schur functors where $S_{\lambda}$ appears with multiplicity $\dim M^{\lambda} \otimes_{C_n} \chi$. If $V$ is purely odd then $V^{\otimes n}$ is modified as an $S_n$-representation by the sign representation which has the effect of replacing every $M^{\lambda}$ with $M^{\lambda^T}$ where $\lambda^T$ is the conjugate partition (I forget what the standard notation for this is), or equivalently replacing every $S_{\lambda}$ with $S_{\lambda^T}$. And the general case is a free product of the purely even and purely odd cases.