I've watched a Let's Play of Saints Row and I enjoyed Saints Row the Third, but neither of them is anywhere near the calibre of the even-numbered games. That said, to get the most out of this one you really should have at least a passing familiarity with all three of the others, since characters and events from those games are referred to repeatedly throughout. The game will still be fun and hilarious either way, but it's better if you already know who the characters are.
The bottom line is, if you haven't played any of the Saints Row games, this is a fantastic game on its own merits and you should get it. If you have played the others then you have even more reason to get this one. If you were disappointed by Saints Row the Third and are worried about being similarly let down by this one, don't be.