Traveling waves of neuronal activity in the cortex have been observed in vivo. These traveling waves have been correlated to various features of observed cortical dynamics, including spike timing variability and correlated fluctuations in neuron membrane potential. Although traveling waves are typically studied as either strictly one-dimensional or two-dimensional excitations, here we investigate the conditions for the existence of quasi-one-dimensional traveling waves that could be sustainable in parts of the brain containing cortical minicolumns. For that, we explore a quasi-one-dimensional network of heterogeneous neurons with a biologically influenced computational model of neuron dynamics and connectivity. We find that background stimulus reliably evokes traveling waves in networks with local connectivity between neurons. We also observe traveling waves in fully connected networks when a model for action potential propagation speed is incorporated. The biological properties of the neurons influence the generation and propagation of the traveling waves. Our quasi-one-dimensional model is not only useful for studying the basic properties of traveling waves in neuronal networks; it also provides a simplified representation of possible wave propagation in columnar or minicolumnar networks found in the cortex.