For those who arrived on this question using Visual Studio 2013 or above, the feature is built right into the program. Just use AltUp and AltDown to move the line with your cursor—or the selected lines—up and down.
If you wish to rebind it in Tools > Options > Environment > Keyboard, the keys are Edit.MoveSelectedLinesUp and Edit.MoveSelectedLinesDown. You may need to remove your new combination from other existing keys first.