I heard back from the Kubota dealer today. Their thought, based upon the information I shared is that I may have broken a clutch fork. I'm not familiar with the inner workings of the tranny, but he made it sound like the clutch fork is inside the beast and is the part that engages and disengages the transmission.
Of course, they couldn't tell until they actually looked at it, but if that's what it ends up being, then they have to open 'er up and get into it. The guy said, "Probably not what you wanted to hear, but we don't have any other ideas at this point."
To all the fellas talking about oil quality and filters, I changed the filters at 250 hours when I bought the tractor and the fluid was good. I topped it off per the manual and dealer when I changed the filters and all has been good. The fluid is STILL clear even with this problem so it looks like I have a pretty significant issue on my hands.
I'll let you all know what the final outcome is when the dealer or other local shop takes a look.