  For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1
\end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0
\end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to
periodic or antiperiodic boundary conditions, we give necessary and sufficient
conditions which guarantee that the system of root functions contains Riesz
bases in $L^2 ([0,\pi], \mathbb{C}^2).$
  In particular, if the potential matrix $v$ is skew-symmetric (i.e.,
$\overline{Q} =-P$), or more generally if $\overline{Q} =t P$ for some real $t
\neq 0,$ then there exists a Riesz basis that consists of root functions of the
operator $L.$
