Fingerprint presentation attack detection (FPAD) is becoming an increasingly challenging problem due to the continuous advancement of attack techniques, which generate `realistic-looking' fake fingerprint presentations. Recently, laser speckle contrast imaging (LSCI) has been introduced as a new sensing modality for FPAD. LSCI has the interesting characteristic of capturing the blood flow under the skin surface. Toward studying the importance and effectiveness of LSCI for FPAD, we conduct a comprehensive study using different patch-based deep neural network architectures. Our studied architectures include 2D and 3D convolutional networks as well as a recurrent network using long short-term memory (LSTM) units. The study demonstrates that strong FPAD performance can be achieved using LSCI. We evaluate the different models over a new large dataset. The dataset consists of 3743 bona fide samples, collected from 335 unique subjects, and 218 presentation attack samples, including six different types of attacks. To examine the effect of changing the training and testing sets, we conduct a 3-fold cross validation evaluation. To examine the effect of the presence of an unseen attack, we apply a leave-one-attack out strategy. The FPAD classification results of the networks, which are separately optimized and tuned for the temporal and spatial patch-sizes, indicate that the best performance is achieved by LSTM.