If germinating outdoors in solo cups, plant them after April 20, when there is no chance of frost for the next 10 days. Check accuweather for about 10 days of 75+ weather. It's OK if the nights get cool, they'll germ. Oh and poke a hole in the bottom of your solo cup for drainage.
Tennessee is a good climate to grow in. I used to live there and grew OD every year. I would plant in late April and harvest sometime between Thanksgiving and Halloween.
I second that as I lived in Nashville for 3 years. Great natural climate for growing both indoors and outdoors.