Android is finally starting to get built-in visual voicemail. The next version of Android, Android M, will include native visual voicemail support in the dialer, rather than requiring each carrier to build its own visual voicemail app. But there's a big caveat: it seems as though each carrier still has to add support for Google's visual voicemail service, and it doesn't appear that most of them will just yet. So far, only T-Mobile and Orange, a carrier in France, are on board, which means that most people are still out of luck. Support for those two carriers should be available when Android M launches this fall — it looks like they're already available in the developer preview as well.
Carriers would probably rather you use their own app
Visual voicemail has been a long time coming on Android. It's been available on the iPhone since the original model launched in 2007, while Android users have largely had to rely on their individual carrier's app or services like Google Voice. Carriers sometimes charge extra for visual voicemail, which may explain why they'd prefer to get customers using their own app; Apple, on the other hand, worked closely with AT&T for its original iPhone launch and was seemingly able to get the feature bundled for nearly everyone. It's not clear how Google has started getting carriers on board, but it's possible that it wanted to build in native visual voice anyway so that Fi subscribers can use it.
The inclusion of visual voicemail in M was revealed in a support ticket, first spotted by Android Police, for the Android developer preview. A Google employee writes that a "basic implementation" of visual voicemail is already present in the developer preview and that it'll be available in the final M release this fall. Voicemail transcriptions are also supported natively, but only if "the carrier chooses to provide them," which none currently do — don't expect to use Google Voice transcriptions either. This is far from an official Google announcement, but it certainly seems like visual voicemail is coming: Android Police also caught the screenshot below, which Google+ user Danny Hollis posted to a Nexus 6 community page.