High-frequency very-long-baseline interferometry (VLBI) observations can now resolve the horizon-scale emission from sources in the immediate vicinity of nearby supermassive black holes. Future space-VLBI observations will access highly lensed features of black hole images -- photon rings -- that will provide particularly sharp probes of strong-field gravity. Focusing on the particular case of the supermassive black hole M87*, our goal is to explore a wide variety of accretion flows onto a Kerr black hole and to understand their corresponding images and visibilities. We are particularly interested in the visibility on baselines to space, which encodes the photon ring shape and whose measurement could provide a stringent test of the Kerr hypothesis. We develop a fully analytical model of stationary, axisymmetric accretion flows with a variable disk thickness and a matter four-velocity that can smoothly interpolate between purely azimuthal rotation and purely radial infall. We then determine the observational appearance of such flows, taking care to include the effects of thermal synchrotron emission and absorption. Our images generically display a "wedding cake" structure composed of discrete, narrow photon rings (n=1,2,...) stacked on top of broader primary emission that surrounds a central brightness depression of model-dependent size. We find that the "black hole shadow" is a model-dependent phenomenon -- even for diffuse, optically thin sources -- and should not be regarded as a generic prediction of general relativity. At 230 GHz, the n=1 ring is always visible, but the n=2 ring is sometimes suppressed due to absorption. At 345 GHz, the medium is optically thinner and the n=2 ring displays clear signatures in both the image and visibility domains, identifying this frequency as more promising for future space-VLBI measurements of the photon ring shape.