The only two that I'm familar with are the 12 valve 5.9 and 8.3 Cummins. I know there are tons more that will run on grease, motor oil, trans fluid, etc.
I'm not 100% but I think the T444e is a 7.3 Powerstroke. I do not know that much about Ford's but I do know guys are running them on grease.
Keep it well filtered and de watered and you "should" be fine.