A .umx file is an Unreal package file containing one or more mod files. The ones I've come across have used either .it (Impulse Tracker mod format) or .s3m (ScreamTracker mod) files. No compression is used, so you can strip out the package header, rename the file and open it in a regular tracker.

ModPlug Tracker (a windows-based multiformat tracker) can open .umx files directly.

Using tracked music makes the music system much more flexible. Mods can incorporate loop points or position jumps within the song, and because mods are based around patterns*, you can have several mixes of a song within one file, set up so that the game can seamlessly segue between mixes. Deus Ex (based on a heavily modified version of the Unreal engine) uses this feature well. Each level has one .umx file for its music, with each file containing four parts, one for when the player is just walking around normally, a second for when the player engages in combat, a third for when the player dies and a fourth part, the purpose of which I haven't found out yet.

The main disadvantage of using tracked music in a game is that it's very hard to use realistic-sounding live instrumentation. Deus Ex, Unreal Tournament and their ilk get around this problem by predominantly using techno music, which is easily constructed using samples, although Deus Ex does contain several good orchestral arrangements.

How to convert a .umx into a mod file:

Open the .umx in a hex editor. Don't use WordPad or other text editors, as they're not really designed to handle binary. This results in you listening to lovely random static. Search for "MTM", "FAR", "SCRM", "MAS_U" or "Extended Module" - they should be a couple of lines from the start of the file.

If you have SCRM, look a couple of characters back and find the hex string "00 4C A1 5B". Delete everything from the start of the file to the end of this string, and save it as *.s3m.

If you have any of the other codes, delete everything from the start of the file to the start of that code (make sure you don't delete the code!) and save the file according to its type:
Code: MTM
Save as: .it
Code: FAR
Save as: .mod (I think...haven't seen this one)
Code: MAS_U
Save as: .ult (again, haven't seen this one)
Code: Extended Module
Save as: .xm

Open up your favourite tracker (I recommend ModPlug Tracker) and enjoy!

A pattern is a 'page' of notes which the tracker plays. Patterns are arranged in a sequence in the tracker. The tracker node explains this much better than I can here.
Kudos to insanefuzzie for pointing out the fact that a .umx is not simply a renamed mod...