Grounding a 3-prong dryer plug
You can retrofit a just ground wire back to the panel. 

They broadly liberalized the rules for this in 2014 to allow this for almost any circuit -- but they had already liberalized the rules for dryer circuits some time prior. In fact, since the washer and dryer plugs both come out of the same service panel, they can share the ground wire.

Grounds don't need to be insulated (must be green or green/yellow if they are) and they don't need to follow the same route as the wiring they are grounding. They do have to be installed using proper wiring methods. Grounds must be large enough for the circuit they are protecting. 

All the circuits out of that panel can share the ground, so it helps to run a "backbone" to the big appliances in #10 or #8, then tie everything else to that in reverse order of size. Splices are allowed in ground wiring, just do it legally in in junction boxes (which means pre-planning where those junction boxes would be useful when laying the backbone.)