InBox has iPhone and web-based clients. I was able to easily "import" my Gmail rules into my account. As the app has continued to improve over the past few months, they have added both the ability to adjust the rules on the fly as you drag messages to folders and the ability to auto-classify messages based on your behavior with them into a small set of standard categories.
Admittedly, it took awhile to wean myself off my old approach, trying for a period of many weeks to use InBox and my primary interface but then using my old MacMail and AppleMail to ensure I didn't miss anything.
Recreating key folders in InBox was easier than Gmail (the rule interface is just simpler).