This paper addresses the hyperspectral image (HSI) denoising problem by using Stein's unbiased risk estimate (SURE) based convolutional neural network (CNN). Conventional deep learning denoising approaches often use supervised methods that minimize a mean-squared error (MSE) by training on noisy-clean image pairs. In contrast, our proposed CNN-based denoiser is unsupervised and only makes use of noisy images. The method uses SURE, which is an unbiased estimator of the MSE, that does not require any information about the clean image. Therefore minimization of the SURE loss function can accurately estimate the clean image only from noisy observation. Experimental results on both simulated and real hyperspectral datasets show that our proposed method outperforms competitive HSI denoising methods.