I'm kind of late to the party, as I'm just watching the show now. I want to say right up front that I do enjoy it, and I'm liking watching it. That being said, I don't agree with the reviewers who believe that it holds up to the original. The introduction of more modern themes and aesthetics bothers me no matter how long I watch the show, and I cringe a little every time a car, motorcycle, or plane appears on screen. For me, at least, it just does not mix at all with the more spiritual, medieval Asian themes of the original series' world. The level of advancement, especially in Republic City, is jarring. When I compare it to some of the impressively gorgeous cities in Last Airbender like Ba Sing Se, Republic City just looks like a garrish New York clone to me.
I actually like Korra as a character, but as the Avatar, it's a little annoying how contrived some of her challenges seem to be. She masters the Avatar State in the very first season, yet somehow struggles to beat some totally ordinary benders in the second. When Aang finally mastered the Avatar State at the very end of his journey, he basically made defeating Firelord Ozai look easy, even at the height of Sozin's Comet. Yet Korra apparently needs a plane to get close enough to a fleet of ships in order to move them out of the way with waterbending? Remember when Aang literally made the seas rise and then fall at the end of Last Airbender using the Avatar State, which Korra theoretically has full access to? Yeah. Don't even get me started on the fact that they've completely tossed all of Korra's past Avatar selves out the window, for... Reasons, I guess.
I'm also not crazy about how each season is its own self-contained storyline, because, as the seasons are shorter anyway than Last Airbender, it feels rushed compared to the three season-long story arc of the original show. I just don't feel like there's enough time given to invest in any of these stories, especially considering Korra saves the world in each season, whereas Aang had to go through a three season journey to do that. What's more, I can't get invested into any villain the same way I did with villains from Last Airbender. Because of the "villain of the week" (as one of the previous reviewers puts it) thing they have going with each season, it's impossible to get invested into a villain the same way I was invested in Zuko, Azula, and, to a lesser extent, Ozai himself (mostly because he was a faceless villain for much of Last Airbender). And there's certainly no character who can undergo a complete character transformation like Zuko.
I guess what I'm trying to say is this; Taken entirely on its own merits, Legend of Korra is a good show that's completely worth watching and enjoying. But as a sequel series to Avatar: The Last Airbender? I find it disappointing and unfaithful.