How common is the idiom “no great shakes” in spoken English, especially among native English speakers? Will I sound natural if I use it in general conversation?
Its use in 21st century TV would indicate that it's not beyond the ken of the average Joe.
From the TV corpus:
I'm such a ninny, I didn't realize it before. I'll never be able to thank you. Oh, no great shakes. (Roots, Episode #1.2, 2016)
He... he always said he was glad that I was born there because, " being a Connecticut Yankee in King Arthur's court is no great shakes. " (Once Upon a Time, An Untold Story, 2016)
But no writer should be judged on their first effort. I mean, the first draft of my novel was no great shakes. - Sure it's better than this. (Murdoch Mysteries, Shipwreck, 2015)
You know, this manager job is no great shakes but it'll keep me here and, I don't know maybe I can go to night school or something. (ER, Drive, 2004)
A google Ngram comparison shows it to be a bit less common presently than run of the mill, but it's been a pretty steady part of the language for 200 years or so.
Looks like you'll be safe using it as long as it fits the context.