As the TCX3 appears to have a standard BIOS and VIA chipset I would not anticipate any problems in running various flavours of Linux
I installed Tiny Core 4.5.2 on the DOM in place of the supplied operating system. This booted and ran quite happily albeit with the processor running at 400MHz rather than its rated 1GHz. Once I got past the BIOS password prompt and loaded the optimised defaults life got better as the processor could run at its rated 1GHz.
Note: I purchased a short 40-pin male-male IDE cable to help with this. That way
I could use a conventional IDE cable with a Compact Flash and the DOM on it. I have
never had any success using the rows of pins that are often sold as male-male adaptors.
The DOM was powered via the cable and I used the spare USB header to provide power
for the Compact Flash card.
I've also run Tiny Core 4.7.6 from a 1GB USB pen drive.