For the 16th C religious discourse, by going back to the primary source material. Most of what has been discussed during the 20th C is still available, and some of it is even available on-line now. Edward VI's journal, for example, is a bit of a hoary nutshell for English historians - they've been through it so many times, there's nothing new for most of them. But I went through it and found seven or eight references to the weather, and, along the way, read the thing. Now, when other historians refer to his journal, I have an opinion.

I do rely explicitly on the historical climatologists, and I wouldn't even begin to know if their data is wrong. Some of their research involves counting tree rings and the width of those tree rings, or analyzing ice-core samples from Greenland. They also incorporate documentary sources into their reconstitutions, though, and I'll be covering some of their ground again because I want to use those same sources for my analysis.