Let $(X_n,\F_n,\P^x)$ be a Marov chain with Markov operator $P$ and put $L\colon=P-1$. Then for all bounded measurable $f:S\rar\R$
$$
M_n^f\colon=f(X_n)-f(X_0)-\sum_{j=0}^{n-1}Lf(X_j)
$$
is a martingale with respect to $\P^x$, i.e. for all $n\in\N_0$: $\E^x(M_{n+1}^f|\F_n)=M_n^f$.
As $X_n$ is Markov we conclude that:
\begin{eqnarray*}
\E^x(M_n^f|\F_{n-1})
&=&\E^x(f(X_n)|\F_{n-1})-f(X_0)-\sum_{j=0}^{n-1}Lf(X_j)\\
&=&Pf(X_{n-1})-f(X_0)-\sum_{j=0}^{n-1}Lf(X_j)\\
&=&f(X_{n-1})-f(X_0)-\sum_{j=0}^{n-2}Lf(X_j)=M_{n-1}^f~.
\end{eqnarray*}