well the MMA fighters do regular training like push ups,sit ups,and all that regular body workouts and if they have there own equipment then they will do there basic training that they do in the gym at home.
I'm not an MMA fighter, but I'm into fitness in general. I use Kettelbells and light dumbbells, but body-weight exercises are my favorite. Do research into variations on push ups, crunches, pull-ups and squats, all using body weight. My new favorite is the burpee.