Mission accomplished! I spent the good part of yesterday afternoon taming our linen closet. I have to say, it looks great. I didn't go out and buy any special organizing stuff for this one. I just worked with a few random boxes and containers I had on hand.
I learned that people do not need an endless supply of cleaning rags (grease-spotted shirts make great rags!). This is all the more true for those who rarely clean. I promptly threw out an entire garbage bag worth of rags.
Originally I thought this closet would take me days to get through. But once the contents were spread out across my bed (I should have thought that through), I realized this was a one-day deal. In and out. And here it is before and after.