The management of water resources must be based on accurate models of the river discharge in the context of the water flow alteration due to anthropic influences and climate change. Therefore, this article addresses the challenge of detecting the best model among three artificial intelligence techniques (AI)—backpropagation neural networks (BPNN), long short-term memory (LSTM), and extreme learning machine (ELM)—for the monthly data series discharge of the Buzău River, in Romania. The models were built for three periods: January 1955–September 2006 (S1 series), January 1955–December 1983 (S2 series), and January 1984–December 2010 (S series). In terms of mean absolute error (MAE), the best performances were those of ELM on both Training and Test sets on S2, with MAETraining = 5.02 and MAETest = 4.01. With respect to MSE, the best was LSTM on the Training set of S2 (MSE = 60.07) and ELM on the Test set of S2 (MSE = 32.21). Accounting for the R2 value, the best model was LSTM on S2 (R2Training = 99.92%, and R2Test = 99.97%). ELM was the fastest, with 0.6996 s, 0.7449 s, and 0.6467 s, on S, S1, and S2, respectively.