I'm probably the minority in this discussion, but here goes.
I like the mechanics of death and dying; however, I would prefer that the Con saving throw be difficult. You have three rounds at 0 hp in which to save of which you have at least a 5% chance of automatically saving and getting 1 hp. I would set the DC at 20 so that pretty much the hardiest fighters (or those who valued Con) get the best chances of surviving (even though those chances are not good) while the others can hope their comrades can get to them in time.
I'm also a big fan of the death spiral. If you're seriously wounded at some threshold (half hp, 25% hp left, or 10% hp left, etc.), then you're taking on penalties to stay conscious while fighting. While fighting is the key. Though to ease book keeping, I would only prefer there be one threshold.
Secondly, dying and coming back from the dead should be a major ordeal. If you die and you're brought back, you lose something more than just getting -4 penalty to rolls until you take long rests to eventually remove them. I'm more in favor of losing levels, particularly if the cost to getting your butt raised is only 500 gp.
I wouldn't want my suggestions to be part of main ruleset, because I know that a lot of players play D&D to play epic heroes and as such dying along the way is not really all that fun; however, for tournament play or gritty games, I'm a big fan where the more stakes on the table makes for a more tense and nerve-wrecking game which is really awesome.