Hi Brady,
Speaking from my personal experience and without knowing your particular climate, I think it is best to harvest roots in autumn after the plant has drawn its vitality back into the root (but before it really is wintertime). What I want to say with that is, that I wouldn´t start harvesting roots before the foliage begins to wither clearly (or in spring when the ground is not frozen anymore, but before the leaves start to grow). In autumn, if you can still identify or mark the place of the root, I would even wait until the foliage is gone as much as possible. This is easy with horseradish since usually you can see a bit of the root still above ground even when the old foliage is totally gone (and often you see already little tips of what will become next year´s foliage). Taste is another indicator, however I could only tell between a root dug in summer and one in autumn/spring. I tried this because I was curious, and in summer the root was not at all spicy and pungent but tasted kind of like cabbage :-)
Burdock is another good one with its distinctive huge leaves. When they become brown and withered they are still there. Only harvest roots of the first year´s growth or beginning of their second year, not the one that have the burrs on them already. However the burrs are medicinal, too, although I have not used them yet, but will collect this year. Dandelion has green leaves in our climate almost all year round, but I learned from folk medicine that you should harvest bitter roots always in spring (however, I can not explain why) so I would go for dandelion and chicory in early spring.