  We use different particular classes of axially symmetric Szekeres
Swiss-cheese models for the study of the apparent dimming of the supernovae of
type Ia. We compare the results with those obtained in the corresponding
Lemaitre-Tolman Swiss-cheese models. Although the quantitative picture is
different the qualitative results are comparable, i.e, one cannot fully explain
the dimming of the supernovae using small scale ~50 Mpc inhomogeneities. To fit
successfully the data we need structures of order of 500 Mpc size or larger.
However, this result might be an artifact due to the use of axial light rays in
axially symmetric models. Anyhow, this work is a first step in trying to use
Szekeres Swiss-cheese models in cosmology and it will be followed by the study
of more physical models with still less symmetry.
