  We investigate spectral properties of Gesztesy-\v{S}eba realizations
D_{X,\alpha} and D_{X,\beta} of the 1-D Dirac differential expression D with
point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty\subset
\mathbb{R}.$ Here $\alpha := \{\alpha_{n}\}_{n=1}^\infty$ and \beta
:=\{\beta_{n}\}_{n=1}^\infty \subset\mathbb{R}.
  The Gesztesy-\v{S}eba realizations $D_{X,\alpha}$ and $D_{X,\beta}$ are the
relativistic counterparts of the corresponding Schr\"odinger operators
$H_{X,\alpha}$ and $H_{X,\beta}$ with $\delta$- and $\delta'$-interactions,
respectively.
  We define the minimal operator D_X as the direct sum of the minimal Dirac
operators on the intervals $(x_{n-1}, x_n)$. Then using the regularization
procedure for direct sum of boundary triplets we construct an appropriate
boundary triplet for the maximal operator $D_X^*$ in the case
$d_*(X):=\inf\{|x_i-x_j| \,, i\not=j\} = 0$. It turns out that the boundary
operators $B_{X,\alpha}$ and $B_{X,\beta}$ parameterizing the realizations
D_{X,\alpha} and D_{X,\beta} are Jacobi matrices. These matrices substantially
differ from the ones appearing in spectral theory of Schr\"odinger operators
with point interactions. We show that certain spectral properties of the
operators $D_{X,\alpha}$ and $D_{X,\beta}$ correlate with the corresponding
spectral properties of the Jacobi matrices $B_{X,\alpha}$ and $B_{X,\beta}$,
respectively. Using this connection we investigate spectral properties
(self-adjointness, discreteness, absolutely continuous and singular spectra) of
Gesztesy--{\vS}eba realizations. Moreover, we investigate the non-relativistic
limit as the velocity of light $c\to\infty$. Most of our results are new even
in the case $d_*(X)> 0.$
