Is this a new thing? I don't recall seeing an option for that in Thunderbird (desktop version by the way; not the mobile / Android version) the last time I looked (~9 months ago).
I don't use Thunderbird myself - was just following the discussion on from the OP who did use it. However I've yet to find a client I like so genuinely interested in any suggestions you might have.
I'm currently using mutt with "getmail" (which does support IDLE), which I can recommend -- it's an excellent client, but only if you're fine with tweaking.
I used TB until two years ago, but I gave up with it's unfixed bugs and quirks. I do prefer graphical clients, but not if they are clunky or buggy.
I used Silpheed and Claws for years, but Silpheed locks (or used to lock) the UI during fetch (unacceptable IMHO) while Claws has some critical bugs in the filter/rule logic that made me lose mail in several occasions by refiling into the wrong folder while processing a lot of messages. If you arent't a heavy filter user you might be fine with it though, I think Claws gets a lot of things right.
KMail wasn't bad when I used it, but it was too long ago to make an honest comment today.
