You can snap to the grid or the cursor by pressing ShiftS, however I'm not sure that is the fastest way in this case.
Method 1
Transform snapping and a guiding plane.
Add a plane underneath the object
Enable Face Snapping:
Press GZ0 with the object selected and your cursor over the plane:
Pros:
- Works in object mode and edit mode
Cons:
Method 2
Positioning and snapping to the 3D cursor with ShiftS.
Snap cursor to face (select face in edit mode and press ShiftS> Cursor to Selected)
Snap the object origin to the 3D cursor (switch to object mode and press CtrlShiftAltC> Origin to Cursor)
Snap the 3D cursor to where you want the face to go (if you want it to go straight down, you can set the Z position of the cursor in 3D view > Tool Shelf (N) > 3D cursor. In the gif I snapped it to the center with ShiftS, however it's slightly fewer clicks to use ShiftC. Note that ShiftC will also center the view between all visible objects)
Snap the object to cursor (ShiftS> Selected to cursor)
Cons:
Requires repositioning of object origin
Only works in object mode (ShiftS works based on the pivot point, so in this case the center of the object would be snapped to the cursor)
Method 3
Snap to grid with ShiftS, then increment snap to align with the gird floor.
Note that Snap to Grid will snap to a 3D grid, not the grid floor (though the grid floor is a 2D representation of the same grid which is used for snapping)
Cons:
- Will snap object to align with grid on X and Y as well.