I managed to figure this out on my own, based on what I had noted above.
Prior to beginning, you must have set up an Environment material to render, or have set up a surrounding scene to capture. Once that is done, you set resolution of the render to a square texture, preferably with a power-of-two size (I set it to 1024).
Once the environment or scene is set up and ready to capture, the first step is to open the camera object's properties, and just below the Perspective/Orthographic/Panoramic selectors, you change the Lens Units from Millimeters to Field of View, and then set the field on the left to 90 degrees.
After that, set the camera's position to the center of the grid (0,0,0 coords), and set its rotation so that it is facing straight forward (rotation 90,0,0). Then render, and save it as the Front texture.
Then rotate 90 degrees counterclockwise, and render as the "Right" texture (even though the camera is facing left). Rotate 90 degrees counterclockwise again, and render that as the "Back" texture. Rotate 90 degrees counterclockwise again, and render that as the "Left" texture.
Rotate 90 degrees counterclockwise again, and then you're back at the starting position. Rotate the camera 90 degrees to point upwards (exact rotation should be 180,0,0), and render that as the "Up" texture. Then rotate the camera 180 degrees to point downwards (exact rotation is 0,0,0) and render that as the "Down" texture.
After that, fire up the Unity3D editor. Import each of the textures (which should be named according to direction). Bring up each texture in the inspector, and make sure that Texture Type is set to Texture, and that Wrap Mode is set to Clamp. Then, create a new Material asset, and in the inspector, set the Shader to "Skybox/6 Sided". Then assign each of the textures to the corresponding slots, and you're done.