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We present theoretical examination and experimental demonstration of locally resonant (LR) phononic plates consisting of a periodic array of beam-like resonators attached to a thin homogeneous plate. Such phononic plates feature unique wave physics due to the coexistence of localized resonance and structural periodicity. We demonstrate that a low-frequency complete band gap for flexural plate waves can be created in the proposed structure owing to the interaction between the localized resonant modes of the beam-like resonators and the flexural wave modes of the host plate. We show that the location and width of the complete band gap can be dramatically tuned by changing the properties of the beam-like resonators. To understand the opening mechanism and evolution behaviour of the complete band gap, some approximate but explicit models are provided and discussed. We further perform experimental measurements of a specimen fabricated by an array of double-stacked aluminum beam-like resonators attached to a thin aluminum plate with 5 cm structure periodicity. The experimental results evidence a complete band gap extending from 465 to 860 Hz, matching well with our theoretical prediction. The LR phononic plates proposed in this work can find potential applications in attenuation of low-frequency mechanical vibrations and insulation of low-frequency audible sound. [ABSTRACT FROM AUTHOR]