As a dispatcher at a worldwide 121 carrier, I use K charts, service volumes (certain 121 flag fuel rules are based on navaid service volumes, while the computer does the math, I have to understand the theory), and area forecasts all the time. If I am working a flight to southern cal, I never go without referring to the SoCal FA to see how much I trust, or dont trust, the TAFs.
I use Lifted Index/K Charts to see how juicy the air is when thunder weather hits - again how much do I trust the TAFs. I've done this long enough to never solely trust a TAF.
There are certain 121 OpSpecs which refer to Class E airspace, and when I was at American Eagle Dispatch back in 2001, we had flights which would depart under VFR all the time and pickup their IFR clearance once airborne (in California), so the clearance from cloud rules were a necessary evil.
There are a lot of places that we fly worldwide where if you didnt have an NDB, you couldnt fly there, so NDB skills are still a gotta have.