So, you want visual voicemail without having to pay the $2.99/month for it?
The answer is already here, it's called Google Voice.

I'm going to show you how to set up Google Voice and visual voicemail on your Verizon Galaxy S 3.

Step 1: Get a Google Voice number. It's free and all you need to do is go to voice.google.com
Follow the steps shown on Google Voice. Yes, you need to authenticate with an existing phone number 
before they give you a Google Voice phone number.

Step 2: Once you have your GV number, I recommend removing any voicemails on your Verizon voicemail mailbox.

Step 3: On Google Voice on your computer click on the gear icon and open settings. Uncheck the boxes for forwarding, 
Click on Groups & Circles. 
Click on the edit button for all contacts and uncheck the boxes labeled Ring my: Mobile (or landline) and Google Talk. 
Click on the edit button for anonymous callers and uncheck those boxes too. The reason for this is to make sure callers
are sent straight to voicemail if you don't pick up.
Check attachments for what to uncheck.

Step 4: Download and install Google Voice from the play store.

Step 5: I like to use the default dialer and my own cell minutes for outgoing calls, but if you want to use your GV number
for outgoing calls you can select that. 
Check attachments for options.

Step 6: Now you are going to change your voicemail to Google Voice in the dialer app. Open the dialer, hit the menu button and tap settings, 
tap voicemail, tap Service and choose Google Voice. You will see a warning, ignore it for now because we'll take care of 
that in the next step.

Step 7: Now you are going to dial some phone numbers to set up the forward on no answer system. Where you see [GV number]
Insert the phone number Google gave you. When you dial these numbers Verizon will automatically hang up on you... jerks.
Numbers to dial:
*71[GV Number]
*90[GV Number]
*92[GV Number]

Step 8: You should now have Google Voice set up so that whenever anybody calls you and it would normally go to Verizon's voicemail
service they will now automatically be forwarded to Google Voice and leave a voicemail there. Go and test it with another
phone.

If for whatever reason you want to remove Google Voice and use Verizon's voicemail just follow these steps.

Step 1: You have already disabled forwarding in Google Voice. This feature sends callers from your GV number to any forwarding phones you set.

Step 2: Open the android dialer, hit the menu button, scroll down to Voicemail, tap it, tap services, and choose your carrier/verizon depending on your rom.

Step 3: Dial *73. Doing this removes the call forwarding settings we did earlier. Dial *86 to make sure you can connect to your voicemail.

SOURCE

Open Google Voice, hit the menu button, go settings, check Voicemail Display, your Google Voicemails will now appear in the call log of your default dialer.