D&D 5th Edition: Importing 13th Age's Quick Rests and Full Heal-Ups

13th Age is a game I would run more if it covered different ground than DnD 5E. My groups, and I can't blame them, basically have said, "Why don't we just play DnD, instead of learning 13th Age?" So I find myself with little outlet for 13th Age at the moment.

However, while reading the 13th Age core rulebook while contemplating using the Escalation die in my Epic Forgotten Realms game later this summer/fall, I was reminded of how 13th Age handles Quick Rests (essentially right after a battle) and Full Heal-Ups (roughly every 4 battles).

It got me thinking about trying that in DnD. My groups play for 2-3 hours tops and we have kept our eye on making sure the different class resources are handled as fairly as possible. While I can't say it's been a huge issue, I wondered about giving this method a try.

Has anyone imported this system into 5E? 

Any experiences, thoughts, or opinions are welcome.