We present a sample of 115 very low optical surface brightness, highly extended, H I-rich galaxies carefully selected from the ALFALFA survey that have similar optical absolute magnitudes, surface brightnesses, and radii to recently discovered "ultra-diffuse" galaxies (UDGs). However, these systems are bluer and have more irregular morphologies than other UDGs, are isolated, and contain significant reservoirs of H I. We find that while these sources have normal star formation rates for H I-selected galaxies of similar stellar mass, they have very low star formation efficiencies. We further present deep optical and H I-synthesis follow-up imaging of three of these H I-bearing ultra-diffuse sources. We measure H I diameters extending to similar to 40 kpc, but note that while all three sources have large H I diameters for their stellar mass, they are consistent with the H I mass-H I radius relation. We further analyze the H I velocity widths and rotation velocities for the unresolved and resolved sources, respectively, and find that the sources appear to inhabit halos of dwarf galaxies. We estimate spin parameters, and suggest that these sources may exist in high spin parameter halos, and as such may be potential H I-rich progenitors to the ultra-diffuse galaxies observed in cluster environments.