I believe that in the sentence Ich danke 'du/dir/dich' that: ich = subject danke = verb du = direct object. Wouldn't that mean that it would be "Ich danke dich"? I hear people say ...
Generally, my understanding of the use of dative and accusative is a bit opaque. I am getting better at deciphering which prepositions call for which case. I have also seen that motion is another ...