This paper proposes a deep sound-field denoiser, a deep neural network (DNN) based denoising of optically measured sound-field images. Sound-field imaging using optical methods has gained considerable attention due to its ability to achieve high-spatial-resolution imaging of acoustic phenomena that conventional acoustic sensors cannot accomplish. However, the optically measured sound-field images are often heavily contaminated by noise because of the low sensitivity of optical interferometric measurements to airborne sound. Here, we propose a DNN-based sound-field denoising method. Time-varying sound-field image sequences are decomposed into harmonic complex-amplitude images by using a time-directional Fourier transform. The complex images are converted into two-channel images consisting of real and imaginary parts and denoised by a nonlinear-activation-free network. The network is trained on a sound-field dataset obtained from numerical acoustic simulations with randomized parameters. We compared the method with conventional ones, such as image filters, a spatiotemporal filter, and other DNN architectures, on numerical and experimental data. The experimental data were measured by parallel phase-shifting interferometry and holographic speckle interferometry. The proposed deep sound-field denoiser significantly outperformed the conventional methods on both the numerical and experimental data. Code is available on GitHub: https://github.com/nttcslab/deep-sound-field-denoiser.