  We address the spectral properties of Mott insulators with orbital degrees of
freedom, and investigate cases where the orbital symmetry leads to Ising-like
superexchange in the orbital sector. The paradigm of a hole propagating by its
coupling to quantum fluctuations, known from the spin t-J model, then no longer
applies. We find instead that when one of the two orbital flavors is immobile,
as in the Falicov-Kimball model, trapped orbital polarons coexist with free
hole propagation emerging from the effective three-site hopping in the regime
of large on-site Coulomb interaction U. The spectral functions are found
analytically in this case within the retraceable path approximation in one and
two dimensions. On the contrary, when both of the orbitals are active, as in
the model for $t_{2g}$ electrons in two dimensions, we find propagating
polarons with incoherent scattering dressing the moving hole and renormalizing
the quasiparticle dispersion. Here, the spectral functions, calculated using
the self-consistent Born approximation, are anisotropic and depend on the
orbital flavor. Unbiased conclusions concerning the spectral properties are
established by comparing the above results for the orbital t-J models with
those obtained using the variational cluster approximation or exact
diagonalization for the corresponding Hubbard models. The present work makes
predictions concerning the essential features of photoemission spectra of
certain fluorides and vanadates.
