Yes, the green ground wire from the switch should be attached via a machine screw (not sheet metal) to the box. Manufacturers provide a tapped hole in the back of the box for that purpose. Use a 10-32 screw (and they sell spiffy little green screws if you want to find those).

Now why the box doesn't have a ground wire is another question with several possible answers. If the box is fed with non-metallic cable (aka Romex) then the bare ground wire of the cable should be attached to the box as described. If the box is fed with metal conduit (EMT) then the conduit functions as the grounding conductor. If there is flexible metal cable, type MC, then the ground wire in the cable should be attached to the box as described. If there is flexible metal cable, type AC, then there is no ground wire. The jacket of cable along with a bonding strip functions as the grounding conductor. The bonding strip does not enter the box but instead is folded back, under the box connector.

Of course, there is the possibility that the last person did something wrong.

Thanks, yes there is always a chance that the last person, which wasn't me, did something wrong. I don't recall at the moment, but I think the circuit was a ground fault fed circuit, in the kitchen. That seems to me to preclude the old fashioned concern about the earth ground.

I still wonder why the plastic encased switch needs a ground for any reason, ground fault or not. I think the box was grounded, not sure it has been several weeks ago and the thought just crossed my mind tonight for some reason. Guess it was my walk through the Lowes electrical section :)

"I still wonder why the plastic encased switch needs a ground for any reason, ground fault or not. "

Is the switch plate metal?
Are the switch plate screws metal?

Would you like the circuit breaker to trip if the dimmer has a fault and energizes those screws or would you prefer to find it when you touch the screw?

Good points, still I wonder how the electricity is going to get from the plastic buried dimming hardware over to the switch plate or screws. Even if it did, those areas are already either grounded, or the attachment of the green wire was to a floating piece of metal.

I'd guess most plates are painted and many are themselves plastic. But, if I had one hand on the sink faucet I would not want the other hand to touch live electricity... that's why it is best to have a ground fault, which the dimmer does not require for installation.

I guess I've got may answer, the reason for the green ground wire is safety, and it has nothing to do with the proper operation - in the since of how the lights operates.

Thanks, not really arguing and I appreciate the confirmation.