From the City of Poughkeepsie, Route 44 east to Route 82 south to Route 343 east is probably the easiest way to do it, but I'm not sure that it's the shortest. Route 55 east to County Route 21 east to Route 22 north is my preferred way of traveling it.
You're probably looking at about 35 to 40 minutes to get across the county. Be advised, though, that as you close in on Dover Plains, there are mountains and the roads start winding. On County Route 21 in particular there are two very sharp turns in succession with speed limits at 15 MPH and below. There is no highway link between the two towns; it's one-lane and/or back roads all the way.
