One of the easiest ways to listen to new voicemails is to get Asterisk/FreePBX to e-mail them to you as an attachment. One downside of this is that if you try to listen to them on your mobile phone they will play via the speaker. This makes it tricky to listen to them discreetly!
For Android there’s a little App called Earpiece that allows you to specify that media is played through the earpiece, rather than the speaker. This is quick to switch on temporarily while listening to the voicemail. It may be hardware dependent, but it works fine on my OnePlus One.