Next time that I DM I think that I'll have it where once you make a death save, your still unconcious until your turn comes around again. Or unless someone else gives you an action.
But that could be because I'm too used to DM's dogpiling every creature possible on the weakest link. I've seen where you wake up with a death save/heal and get gang banged before your turn rolls around happen way too often for my tastes.
If you're playing with a DM like that, don't heal someone who's just going to get dog-piled. Tactically, it's better to let them roll a few death saves (hey, they can fail two and be no worse for wear in 5 minutes!) and use the heal once the rest of the party is in a position to protect them.