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In recent years, neural network based image denoising approaches have revolutionized the analysis of biomedical microscopy data. Self-supervised methods, such as Noise2Void (N2V), are applicable to virtually all noisy datasets, even without dedicated training data being available. Arguably, this facilitated the fast and widespread adoption of N2V throughout the life sciences. Unfortunately, the blind-spot training underlying N2V can lead to rather visible checkerboard artifacts, thereby reducing the quality of final predictions considerably. In this work, we present two modifications to the vanilla N2V setup that both help to reduce the unwanted artifacts considerably. Firstly, we propose a modified network architecture, i.e., using BlurPool instead of MaxPool layers throughout the used U-Net, rolling back the residual U-Net to a non-residual U-Net, and eliminating the skip connections at the uppermost U-Net level. Additionally, we propose new replacement strategies to determine the pixel intensity values that fill in the elected blind-spot pixels. We validate our modifications on a range of microscopy and natural image data. Based on added synthetic noise from multiple noise types and at varying amplitudes, we show that both proposed modifications push the current state-of-the-art for fully self-supervised image denoising.