Don't get me wrong - David Tennant gave a great performance. He was funny, energetic, emotional, and very attractive. He had great companions, perhaps the best being Donna; a surprisingly good performance by the comedian Catherine Tate. He had superb villains to deal with, including the introduction of the Weeping Angels, and an utterly mad Master.
But for me the question is who was the character that Tennant was playing, and I just can't see it as being the Doctor.
The Doctor is an ancient alien from a race nearly as old as time. His race doesn't share human concerns, and live complex lives, regenerating into new bodies with new personalities. His mind isn't anything like a human mind - his brain is much more powerful and he has telepathic abilities. His mind also is connected to time and space in a way utterly different from that of a human (as described in "The Parting of the Ways").
The Doctor is basically good - he rebelled against the Time Lords because he would not stand back and watch the suffering of other species. He decided to intervene, to help others. But his ways aren't always clear to humans. The Doctor is the friend who we need but we never really understand.
David Tennant played the part of a human called "Doctor". There was little of the alien about him. He loved, he laughed, he suffered and he triumphed as a human would. Worst of all, he seemed to forget that he was a Time Lord when his body finally met it end, with a childish "I don't want to go!". (Compare that to the wonderfully dignified endings of Tom Baker's Doctor or Matt Smith's).
Christopher Eccleston's Doctor was strange enough - an excess of vivaciousness hiding the deep scars of the Time War. Matt Smith's Doctor was manic, very much like the wonderful Patrick Troughton's interpretation of the character. Peter Capaldi's Doctor is the most alien of any; mercurial, eccentric, and having difficulty understanding humans at all, but the core of the Doctor is still there, as it should be.
So, this is my position: David Tennant's Doctor was a great character, but it wasn't the Doctor.