Turn off the furnace. Go to the highest vent in the house. Spread newspapers out in front of it.
Fill a bucket with warm water and a little liquid household cleanser. Dampen a sponge in it. Sponge off the grill, getting off all the dust and dirt.
Remove the vent grill with your screwdriver. Sponge off the back side of it. Set it on the newspapers to dry.
Sponge off the inside of the open vent area, wiping down the walls all along the opening and in as far as you can reach.
Assemble your vacuum hose and brush. Wrap a piece of tape around each point where the pieces connect, so it won't fall apart inside the vent.
Vacuum out the inside of the vent for as far as you can reach, running the brush against the sides. Re-install the vent grill.
Repeat the process for each vent grill in the house. Replace the old furnace filter with the new one. Turn the furnace.