You can open your RC desktop app, select edit button and select all the message you need to delete at once:

My screen looks different, I do not see the pen...any suggestions?

Good o' @Anirban is showing the classing RC app. However, you, @Deb Hendricks, are in the new app. RC won't let you login to the old app anymore. So if you don't see the edit icon, you probably can't do it. I'll come back with the answer if I find one. :-(

I just came across this thread as I was looking for this feature as well. I have also voted on the feature request. If anyone else comes across this, here is the link to upvote the feature request:

https://ideas.ringcentral.com/ideas/CUSTCOM-I-2363

In the meantime, this can be done in the online portal.

Almost 2 yrs and this still hasn't been done ?