"I'm not going to worry."
Translation:No voy a preocuparme.
What is up with the "me" in "preocupar'me'"?
The me in preocuparme is because the verb is reflexive in Spanish. We more or less say to worry oneself.
Why is "Yo no voy a me preocupo" wrong?
Because the Spanish structure for the close future (to be going to + -ing form in English) is ir a + Infinitivo. Thus:
ir + a + Infinitivo = No voy a preocuparme