What's wrong with tab bars? I mean, most iOS apps use it and if you need more navigation items, you devote one tab for "More". the Android version of Instagram does this the right way.
The difference is convenience. You don't have to reposition your fingers to access the menu. IMO, the best example of this UI pattern is Android Browser (at least on tablets, don't have an Android phone yet) - you slide your finger from either side of the screen and get a circle menu, which looks rad and is really much more convenient to use.
Pro: tab bars are modeless.
Con: less content, more chrome.
The tab bar is pretty poor compared to this. You get 4 things and a "More" button, but what if you commonly need to access more than 4 things?