Although there is skepticism about the likelihood of predictive success, research on the prediction of an earthquake through precursory changes in natural parameters, including groundwater, has continued for decades. One of the promising precursors is the changes in groundwater, i.e., the level and composition of groundwater, and the monitoring networks are currently operated to observe earthquake-related changes in several countries situated at the seismically active zone. In Korea, the seismic hazards had not been significantly considered for decades since the seismic activity was relatively low; however, the public demands on the management and prediction of earthquakes were raised by two moderate-size earthquakes which occurred in 2016 and 2017. Since then, a number of studies that were initiated in Korea, including this study to establish a pilot-scale groundwater-monitoring network, consisted of seven stations. The network is aimed at studying earthquake-related groundwater changes in the areas with relatively high potentials for earthquakes. Our study identified a potential precursory change in water levels at one particular station between 2018 and 2019. The observed data showed that most monitoring stations are sufficiently isolated from the diurnal natural/artificial activities and a potential precursory change of water level was observed at one station in 2018. However, to relate these abnormal changes to the earthquake, continuous monitoring and analysis are required as well as the aid of other precursors including seismicity and geodetic data.