In the past, I've considered the option of having ring cooldowns affect all of a player's rings - normally, in the hopes of eliminating Stamina, which I consider a pretty worthless system. However, that change has some obvious issues, in that it orients the game around waiting - you would spend a lot of time 'doing nothing', and you'd certainly feel it.
However, what I was considering recently was something a little gentler - specifically, simply preventing players from using a ring while they're still 'using' another; requiring that the animation from one ring 'resolve' before you can initiate a new one. An immediate concern, of course, would be that some rings have animations longer than is strictly necessary - however, if we're defining 'resolution' from within the animation itself (rather than simply assuming it happens at the end), we should be able to place where it makes the most logical sense, on a ring-by-ring basis (i.e. after a player fires Hunter's Bow, not necessarily after it hits).
Now, I suppose the question would be why. It adds an additional 'cost' to using active rings, generating a functional barrier against SAD-style sets - by restricting the number of rings that can actually be used in a given span of time, the game allows the first rings to cooldown, making it impossible to constantly use several rings as they cool down; this implicitly teaches players that the 'leftover' slots in their build should be non-active rings (like buffs), in a way that simply running them out of Stamina never did.
Secondly, making active rings 'cost' more necessarily pushes the focus away from them; currently, the game is very hack-and-slash, and oriented around spamming the same effects constantly, regardless of situational concerns. By making spamming rings more difficult, you require players to actually think about what they're using; and, interestingly, you actually give them the time do so. Talk about self-supportive mechanics.
I'll admit I have a personal bias against 'spamming' attacks - I think the game really should be drawing attention to individual rings, rather than simply letting players mash them together into a damage-swarm. Of course, this would be best if rings actually served unique functions, rather than simply being a gradually decreasing scale of damage-efficiecncy - but that's an old topic. The biggest issue I see with this suggestion is that it seems to unfairly punish fast rings - a player with Heavy Water Balloon and Fire Rain won't mind sitting through animations nearly as much as a player using Mantis and Guns, Guns, Guns. Additionally, it can cause a ring's cooldown to belie its actual 'speed' - depending on the length of their animations, a player with 2-second, 3-second, and 4-second rings might use all three equally often, particularly if he favours the higher-cooldown rings over lower-cooldown ones. In that manner, it adds something of an additional complexity to assembling one's build - however, I think it's a grokkable sort of complexity (players can figure out easily that their 'fast' rings get slower when they can't always use them), and I don't believe setbuilding is considered prohibitively complex already...