The determination of temporal and spatial correlations in neuronal activity is one of the most important neurophysiological tools to gain insight into the mechanisms of information processing in the brain. Its interpretation is complicated by the difficulty of disambiguating the effects of architecture, single-neuron properties, and network dynamics. We present a theory that describes the contribution of the network dynamics in a network of "spiking" neurons. For a simple neuron model including refractory properties, we calculate the temporal cross-correlations in a completely homogeneous, excitatory, fully connected network in a stable, stationary state, for stochastic dynamics in both discrete and continuous time. We show that even for this simple network architecture, the cross-correlations exhibit a large variety of qualitatively different properties, strongly dependent on the level of noise, the decay constant of the refractory function, and the network activity. At the critical point, the cross-correlations oscillate with a frequency that depends on the refractory properties or decay exponentially with a diverging damping constant (for "weak" refractory properties). We also investigate the effect of the synaptic time constants. It is shown that these time constants may, apart from their influence on the asymmetric peak arising from the direct synaptic connection, also affect the long-term properties of the cross-correlations.