If you, like me, only are having trouble getting the No Death
Start your next hospital, by building absolutely nothing the first year and send every patient home as soon as they arrive.
When the first year have passed, you'll get the No Death
-award. No patients == no deaths :)
Now build your hospital like you normally would and earn the remaining awards.
If the cheevo doesn't pop, after you have gotten at least one of all awards; press
and go the overview tab and scroll to the awards and then it should pop :)