  Given a sequence of polynomials $(p_n)_n$, an algebra of operators
$\mathcal{A}$ acting in the linear space of polynomials and an operator $D_p\in
\mathcal{A}$ with $D_p(p_n)=np_n$, we form a new sequence of polynomials
$(q_n)_n$ by considering a linear combination of $m$ consecutive $p_n$:
$q_n=p_n+\sum_{j=1}^m\beta_{n,j}p_{n-j}$. Using the concept of
$\mathcal{D}$-operator, we determine the structure of the sequences
$\beta_{n,j}, j=1,\ldots,m,$ in order that the polynomials $(q_n)_n$ are common
eigenfunctions of an operator in the algebra $\mathcal{A}$. As an application,
from the classical discrete families of Charlier, Meixner and Krawtchouk we
construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of
higher order difference operators.
