Aha! Here's an episode from the David Feldman Radio Show which was recorded about this time last year. I always thought she really sounded like the teacher from "Frosty the Snowman", Magica de Spell, or Grammi Gummi, three voices which I love. But here's a sampling of her actual voice:
https://www.youtube.com/watch?v=hoNZrH9TBdU
Enjoy!

Heya Chance and all. . .
As I wrote in another thread here, I am June's cousin.
I had the great fortune to speak with her over the phone several days ago (and to her younger sister a few days after that), her publisher put me in contact as I had family details, which matched up with what June had been telling him in the bits of her autobiography (a work in progress) she has sent him.
She and her sister sound very much alike; a very much more mature and even toned Rocky, Natasha, Nell. . .June is just the most gracious, engaging person.
She is totally aware of the joy she's brought to millions over the years and is wholly appreciative that we, her fans, have had that enjoyment, plus she had some family info I didn't, as did I for her.
I am still floating on air!
May she go on as long as life continues to interest her.
Steven

Sorry if this reply is a little late, but if you search her name on YouTube you can find clips of her giving interviews in her regular voice.
"I think Homo sapiens and Lebanese should be left to their own devices."-Fred Scuttle

Like a more femenine version of Rockyor you might say, a less high pitched version of Nell from Dudley Do-Right or Marigold from Tom Slick (Tom Slick of George of the Jungle fame).
Brian

Kinda like a mix of a dialed down Rocky mixed with a younger Granny. All the DVD copies of The Grinch from 2001-2005 if i'm not mistaken have a commentary track with her and some one else. It was dropped for the new edition but it is out there.