Here's our starting cube:
And after I add some aligned dimensions...
But clearly that's not quite right. The dimension line is parallel to the object being measured, as it should be, but we'd also like the extension lines to be parallel with the edges of the cube.
To accomplish this, I'll use the Oblique option of the DIMEDIT command. Obliquing a dimension angles the extension lines away from their normal alignment.You can type it on the command line, or find it on the flyout of the Dimensions panel on the ribbon.
After you select your dimension to oblique, you need to enter an obliquing angle. This angle represents the new, absolute angle of the extension lines -- not a delta from whatever the current angle is.
So to get my cube dimensioned correctly, I obliqued two dimensions to 30 degrees and the third to 150 degrees.