In synchrotron tomography, experimental constraints can impose dose and time constraints that lead to noise that carries over into the reconstructed images. Convolutional neural networks (CNNs) have rapidly gained popularity as a powerful tool for removing noise from reconstructed images. However, training CNNs typically requires collecting a dataset of paired noisy and high-quality measurements, which is a major obstacle to their use in practice. To circumvent this problem, we have proposed a method for training deep convolutional neural networks to denoise tomographic reconstructions that does not require any noise-free data. We present results on challenging dynamic micro-tomography and X-ray diffraction computed tomography datasets. In addition, we discuss challenges and provide an outlook on future work.