We present the results from the neutral hydrogen (H i) follow-up survey of 378 optically detected ultradiffuse galaxy (UDG) candidates from the Systematically Measuring Ultra-Diffuse Galaxies survey using the Robert C. Byrd Green Bank Telescope. We detect H i in 110 targets and determine 37 to be UDGs and 73 to be low-surface-brightness (LSB) dwarfs based on their effective radii and central surface brightnesses. In line with previous studies, we find that: (i) our H i detections have on average bluer g − r colors and more irregular morphologies than our H i nondetections, (ii) our H i detections populate the tail end of the star-forming main sequence from the ALFALFA catalog with marginally lower specific star formation rates, and (iii) H i detections are mostly in relatively isolated (i.e., field) environments, while most nondetections have at least one nearby neighbor in projection. We find that the H i mass to stellar mass ratios (i.e., gas richnesses) scale with the physical sizes for UDGs and LSB dwarfs alike, suggesting that mechanisms other than bursty star formation feedback may be at play for UDGs. However, we find a stronger trend between gas richnesses and physical sizes if we define UDGs using their effective surface brightness instead of their central surface brightness. We are in the process of using this unprecedented sample of UDG candidates to carry out detailed follow-up studies (i.e., star formation and environmental analysis, comparisons to simulations) and are obtaining resolved H i observations for several of them.