I had a DCT-6200 early in 2007 and improved the Firewire control of the box by MythTV to the point where I was happy with it. But this code is only in MythTV SVN, not in trunk. We're hoping to do a MythTV release in Feb/March, and I'll generate LinuxMCE packages for this shortly after the release.
The MythTV 0.20.2+fixes in LinuxMCE 710 can control a DCT-6200 via firewire, but the setup is problematic and sometimes needs to be reconfigured after the DCT-6200 is rebooted or another device is added to the bus (I used a Firewire hard drive to test the fix which is in the next MythTV release.)
For now, I would recommend Thom's solution of using IR control.