iPhone Voicemail

My iPhone's voicemail can receive Voicemail messages but I cannot access them.

It asks me for my voicemail access code but doesn't recognise any it might be. I have tried to reset voicemail password but it always says "Voicemail unavailable. Unable to connect. Try again later." 

I have reset network settings, turned the iPhone off and on, fiddled with Airplane Mode on and off, and everything else I have read on the Internet!

I called my carrier and they said that voicemail is turned on, and indeed it is as people can leave messages.

Is the voicemail access code something I can change from my phone or is that something I need to set with the carrier? And if so, then what is voicemail password?

Thanks!