Early detection is essential to reduce cancer mortality. Oral cancer could be subject to screening programs (similar as for cervical cancer) by collecting Pap smear samples at any dentist visit. However, manual analysis of the resulting massive amount of data is prohibitively costly. Convolutional neural networks (CNNs) have shown promising results in discriminating between cancerous and non-cancerous cells, which enables efficient automated processing of cancer screening data. We investigate different CNN architectures which explicitly aim to utilize texture information, for cytological cancer classification, motivated by studies showing that chromatin texture is among the most important discriminative features for that purpose. Results show that CNN classifiers inspired by Local Binary Patterns (LBPs) achieve better performance than general purpose CNNs. This holds also when different levels of general data augmentation, as well as pre-training, are considered.