For snobs like me who wants to use keyboard only:

1. Select the cells you want to move (doesn't necessarily need to be a complete row for this)
2. Move to the cell immediately below the one you want to move the content to
3. Press ctrl-+

For snobs like me who want to use keyboard only:

1. Select the cells you want to move (rows or columns, or any rectangle of cells, but the selection must be contiguous)
2. Press Ctrl+x (Ctrl+c would let you make two copies.)
3. Move to or select the cell immediately after where you want to move the content (below it, or to the right.) This cell will not move.
4. Press Ctrl++. This will shift cells down or to the right. (Ctrl+v would overwrite cells instead.)

On the Mac, all of these shortcuts work the same with either Ctrl or ⌘.