Linen closets are typically going to be mostly shelves. Typically you need bigger shelves, wider shelves, or deeper shelves depending on what kinds of linens and things you are storing in there. I think the important thing is to keep the items that you use the most at a comfortable level of reach. So waist high to head height is the area that is easiest for us to reach. This is where you would keep your towels, blankets that you change a lot, and sheets that you are changing all the time. Keep these on those middle shelves. Other items that are less important, like table cloths that you use every once in while, or big blankets that are only used at certain times, or pillows or things that are stored only for when guests come, can be stored up high.
