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Multi-pitch estimation is critical in many applications, including computational auditory scene analysis (CASA), speech enhancement/separation and mixed speech analysis; however, despite much effort, it remains a challenging problem. This paper uses the PEFAC algorithm to extract features and proposes the use of recurrent neural networks with bidirectional Long Short-Term Memory (RNN-BLSTM) to model the two pitch contours of a mixture of two speech signals. Compared with feed-forward deep neural networks (DNN), which are trained on static frame-level acoustic features, RNN-BLSTM is trained on sequential frame-level features and has more power to learn pitch contour temporal dynamics. The results of evaluations using a speech dataset containing mixtures of two speech signals demonstrate that RNN-BLSTM can substantially outperform DNN in multi-pitch estimation of mixed speech signals.