I've never really known what dreich means, but then as someone living in the south of England I've only ever come across it in books, so it's never mattered that much.
I've always had a vague idea it means something like drily obstinate, probably because it sounds a bit like dry* and seems, from the occasional contexts in which I've come across it, to be something inconveniently egotistical.
I've probably been quite wrong (I'm going to look dreich up in a moment) but that's been my longstanding position on the word.
I also assume it's Scottish.
The really appalling thing is that a good chunk of my vocabulary exists in the same shadowy valley of ignorance. (As, probably, does yours.)
Still, that's possibly better than the sunlit uplands of confident ignorance.
Anyway. Dreich. Hang on...
Ah yes, Here we are.
Well, it is Scots, and there is perhaps a subtle dryness to it, but obstinacy...not really.
Dreich means dreary.
Still, dreariness is frustrating, just like obstinacy.
Thing Not To Be Today: dreich. (Or, apparently, dreigh). This word comes from the Middle English dreig, enduring, from some Old English word which looked something like drēog.
*Dreich doesn't actually sounds like dry at all: you say it dree- and then the sound you get at the end of loch. But then, as I've said, I've never heard anyone say it.