Tissue-processing containers are disclosed for facilitating multistep processing of tissue. Also disclosed are systems incorporating the tissue-processing containers that include shakers, incubators, controllers, and pumps. Multistep methods are disclosed for processing tissues using the tissue-processing containers. The tissue-processing containers are specially designed to achieve efficiencies in automated tissue processing, exposure of tissues to processing media, and multistep tissue processing protocols. The described tissue-processing containers comprise a cup-like cavity comprising a middle hole covered by a mesh screen, troughs for separate processing steps, and are stackable in a substantially airtight and substantially watertight manner. Tissues for processing using the described tissue-processing containers include nerve tissue. Also described herein is a transport housing for the transport of stacks of tissue-processing containers, and which may directly incubate those containers, or which may be placed inside a separate incubator for incubating the containers.