  We study effects of inhomogeneities on distance measures in an exact
relativistic Swiss-cheese model of the universe, focusing on the distance
modulus. The model has LCDM background dynamics, and the `holes' are
non-symmetric structures described by the Szekeres metric. The Szekeres exact
solution of Einstein's equations, which is inhomogeneous and anisotropic,
allows us to capture potentially relevant effects on light propagation due to
nontrivial evolution of structures in an exact framework. Light beams
traversing a single Szekeres structure in different ways can experience either
magnification or demagnification, depending on the particular path, and we
explore a small additional effect due time evolution of the structures. We
study the probability distributions of $\Delta\mu=\mu_{LCDM}-\mu_{SC}$ for
sources at different redshifts in various Swiss-cheese constructions, where the
light beams travel through a large number of randomly oriented holes with
random impact parameters. We find for $\Delta\mu$ the dispersions $0.004\le
\sigma_{\Delta\mu} \le 0.008$ mag for sources with redshifts $1.0\le z \le
1.5$, which are smaller than the intrinsic dispersion of, for example,
magnitudes of type Ia supernovae. The shapes of the distributions we obtain for
our Swiss-cheese constructions are peculiar in the sense that they are not
consistently skewed toward the demagnification side, as they are in analyses of
lensing in cosmological simulations, reflecting a limitation of these
constructions. This could be the result of requiring the continuity of
Einstein's equations throughout the overall spacetime patchwork, which imposes
the condition that compensating overdense shells must accompany the underdense
void regions in the holes. The possibility to explore other uses of these
constructions that could circumvent this limitation and lead to different
statistics remains open. (Abridged)
