How to get a Ring of Royal Grandeur?
Credit >> Junger. For details read here: http://us.battle.net/d3/en/forum/topic/12675389719

Junger Rules:

- Do not open ANY chests (including cursed shrine or cleansing chest events), unless you have to for the bounty completion (Matriarch’s Bones, Jar of Souls, Garrach the Afflicted)

- Do not pop any lootable bodies, or objects (tiles)

- Try not to pop any breakables, this is hard to control with Thunderfury and certain classes/specs, but do your best.

- Try to do at least 1 hour of bounty farming to utilize the Pity Timer effect

- Optional – MF Gear, if it doesn’t impede your clear time, why not.

- Optional – I like to open them in Act 1 on T6, doesn’t hurt right.

TL;DR - Clear bounties without opening chests, bodies, breakables, and you should get a legendary in your caches after 1 hour or so of doing A1 bounties that may be a RoRG. If you get legendary drops while clearing, you probably won't have a legendary in your caches.

I personally tested this. And it worked.

Without jungle rule, averagely I got a ring/ammy/wep from every 30 cache (I could not even get 1 from 50 cache, sometimes).
With jungle rule, averagely I get a ring/ammy/wep from every 5 cache.
The drop rate does increase with jungle rule, at least based on my routine farming for rift keys.

What I do is,
1. I don't open chest, let it be big or small.
2. I don't touch the corpse.
3. I solve curst chest quest.
4. I kill elites and gobs as many as possible (yes, Nemesis bracers are used on every shrine).

been doing this since last 2 month.

and yes, it does works. Higher chance. I talked about this on LYN clan before xD