I think I had read somewhere around the 2000 era?
Edit--- Here is the link for the WSMs but not sure about other models.. http://www.virtualweberbullet.com/woodenhandle.html
The Weber search on the web claims it was in the late 90s on the others...
I'm pretty sure the switch was with the 1999 models. I bought my 1998 tagged (EZ) Weber 22.5" because of the wood handles and kept it in the box until I assembled it in 2007. I saw the new models that were due to come out (1999) had plastic handles and I didn't want to go that route. I've since paid the price with my SJP, WSM and Gas Go Anywhere...
Dennis, in the 1960's, Weber used rivets to fasten the wooden pieces to the metal handle. If it's older than 1979, you won't find a date code on any of the vent dampers.