  Light propagation in two Swiss cheese models based on anisotropic Szekeres
structures is studied and compared with light propagation in Swiss cheese
models based on the Szekeres models' underlying LTB models. The study shows
that the anisotropy of the Szekeres models has only a small effect on
quantities such as redshift-distance relations, projected shear and expansion
rate along individual light rays. \newline\indent The average angular diameter
distance to the last scattering surface is computed for each model. Contrary to
earlier studies, the results obtained here are (mostly) in agreement with
perturbative results. In particular, a small negative shift, $\delta
D_A:=\frac{D_A-D_{A,bg}}{D_{A,bg}}$, in the angular diameter distance is
obtained upon line-of-sight averaging in three of the four models. The results
are, however, not statistically significant. In the fourth model, there is a
small positive shift which has an especially small statistical significance.
The line-of-sight averaged inverse magnification at $z = 1100$ is consistent
with $1$ to a high level of confidence for all models, indicating that the area
of the surface corresponding to $z = 1100$ is close to that of the background.
