Google Voice. You don't have to listen to messages at all if you don't want to, just clear the notification indicator and let it sit in your inbox forever or delete it from the GV app or you PC.
Set up an account (no invites needed anymore), get a new phone number (in case you decide to fully use it) and set it up for voicemail. People still call your regular number but you don't use Verizon, you use Google.
I love me some GV, I use it for texting (free!) free calls (friends/family) and voicemail, it also comes in handy for when you don't want to give someone your 'real' number. It has a lot of uses but voicemail is great!