Amazon is now making it easier to delete Alexa recordings with new voice commands. Now if you say, "Alexa, delete everything I said today," your "assistant" will clear any recordings it stores on the Amazon servers. Amazon says that another command, "Alexa, delete what I just said," will work as well.
Before this point if you wanted to delete your voice recordings you would either have to go into the app to delete Alexa recordings one by one or you would have to go through Amazon's website to delete every recording at once. Amazon stores every request and command you make towards your Alexa device to apparently help improve voice recognition services and other features.
This feature may not thouroughly effect everyone's concerns with privacy but it does start the process of deleting your voice data a bit easier.