Morkus from Orkus
In essence they can. With a word Zeus fries mortals with lightning. With a word Apollo incinerates them with the heat of the sun(Flamestrike or something). With a word Hades kills them with death. You could even come up with something that kills by overpowering love for Aphrodite.I was kinda think the ability to Kill (with a word) and Resurrect mortals (to a limited extent) was something that should be available to all gods. But maybe I need to rethink that.
Returning to life is not as natural for them. Orpheus has to go to Hades to have his wife returned alive. His father Apollo can't do it. Still, I don't see it being a big deal to give all gods the power to return mortals to life. If you want to do it that way, demigods could have raise dead, lesser gods could have resurrection, and greater gods could have true resurrection.