We study two Kitaev-Heisenberg t-J-like models on a honeycomb lattice, focusing on the zigzag magnetic phase of Na(2)IrO(3), and investigate hole motion by exact diagonalization and variational methods. The spectral functions are quite distinct from those of cuprates and are dominated by large incoherent spectral weight at high energy, almost independent of the microscopic parameters-a universal and generic feature for zigzag magnetic correlations. We explain why quasiparticles at low energy are strongly suppressed in the photoemission spectra and determine an analog of a pseudogap. We point out that the qualitative features of the predominantly incoherent spectra obtained within the two different models for the zigzag phase are similar, and they have a remarkable similarity to recently reported angular resolved photoemission spectra for Na(2)IrO(3).