While it's true much of what you do in Dragon Age II isn't directly connected to the plot of Dragon Age: Origins, the game assumes much aboutyou are familiar with the setting established in Dragon Age: Origins. The assumptions are enough suchDragon Age: Origins so much so that it completely glosses over a ton of exposition which serves to give meaning to the motivations of the main character and the overall story.
That is, if you don't play Dragon Age: Origins, you're going to find yourself asking "Who cares?" more often than not while the plot unfolds. Dragon Age II will give you the CliffsNotes version of a plot element, but the background behind that plot element is fully explored in Dragon Age: Origins.
Additionally, there's a ton of dialogue points in sidequests and with companions which make reference to the events of Dragon Age: Origins that'll be lost on the person who doesn't play it.
But perhaps the most important reason to play Dragon Age: Origins before Dragon Age II is that it's the better game. Dragon Age II has better gameplay and a much improved user interface, but it's not the Godfather II of the series. You'd be doing yourself a disservice by not playing Dragon Age: Origins in order to play Dragon Age II exclusively.