A discrete denoising algorithm estimates the input sequence to a discrete memoryless channel (DMC) based on the observation of the entire output sequence. For the case in which the DMC is known and the quality of the reconstruction is evaluated with a given single-letter fidelity criterion, we propose a discrete denoising algorithm that does not assume knowledge of statistical properties of the input sequence. Yet, the algorithm is universal in the sense of asymptotically performing as well as the optimum denoiser that knows the input sequence distribution, which is only assumed to be stationary. Moreover, the algorithm is universal also in a semi-stochastic setting, in which the input is an individual sequence, and the randomness is due solely to the channel noise. The proposed denoising algorithm is practical, requiring a linear number of register-level operations and sublinear working storage size relative to the input data length.