I'm adding an update to the Back to Basics section of my D&D 5e House Rules that goes a little something like this:
When someone hits your flesh with a critical hit, you start bleeding. Note the highest die size among the dice used for the damage roll that caused the bleeding. At the beginning of each of your turns, you roll one die of that same size - called the bleed die - and automatically take that much damage. This effect ends when you regain any amount of hit points, or on a successful Medicine roll with a DC Equal to 6 + the size of the bleed die. Bleeding does not stack, but additional damage to your flesh can cause your bleed die to increase in size. Outside of combat, bleed damage continues to take effect once per exploration turn.
I like this little touch because characters aren't going to be able to take wounds and forget them; their wounds are something they have to carry with them, a resource they have to manage during times when they'd ordinarily forget about it.
This can be adapted to the core rules by just having a character bleed on a critical hit.