weeding the new-to-me raspberry patch. I've pulled the gray and obviously dead canes. Do I leave the non-bearing canes that are still supple but have no leaves, or should I leave them alone?

is that, should I leave them alone or should I leave them alone???  sorry, I think I know what you mean.  I'm a little confused by "non-bearing canes that are still supple but have no leaves" -- do you mean they're so little they haven't broken bud yet?  If you have no green buds, the cane is probably dead from some cause...

For this learning season, I would definitely say, LEAVE THEM ALONE.  They are where the fruit will be next year!  AFTER the current crop is finished, thin the shoots, to the biggest, healthiest looking 3.  AND the old canes should come out as soon as their crop is done, so that all the energy goes into the new canes.

Also, your current bare canes may fruit later this summer, if you have a two-crop variety!  This spring's shoots make fruit this fall; this fall's shoots make fruit next spring.  If you get two crops, do the thinning twice, once after each crop.