Between Elkton and Harrisonburg there are only a few dirt roads, none of which are challenging. If you are looking for a short detour, I would reccomend 340 South to Captain Yancey Rd., Right at Island Ford Rd then Left after the bridge onto Old Island Ford Rd aka Power Dam Rd(Dirt) along the river. This eventually will spit you out on 996 (McGaheysville Rd) or you can take a left on Three Springs Rd past the water treatment plant and then a Right on Pineville and Left on McGhaeysville Rd. Take a left at McGhaeysville Rd and this will bring you back up to Rt 33. Left to go to Harrisonburg. There was an appple tree at the old Middle School on 996 that had a great bounty of fruit. Probably all gone now, but I was filling up backpacks full of tasty free apples through the fall.
This is just a short detour through some countryside but by no means a "dirt" Route.
A nice ride to the East is to go down 340 and turn onto Ore Bank Rd or Raynes Bank Rd. These will take you South and eventually to Browns Gap aka Madison Run Fire Road. These are mostly dirt and the site of the Mt. Vernon Forge. Jubal Early fortified himself here during the Civil War and whole armies were once transported through this path. Browns Gap Rd heading East then turn Right onto Port Republic Rd. This once again crosses 340 and continues back through the town of Port Republic, past Cross Keys and to Harrisonburg.
Not sure if you have a lunch destination picked out but the Little Grill has some healthy options and a great "local" burger.
This all may have been way out of line but I just thought I'd share some nice routes.