This actually isn't that hard to do. Want to play as Korea, but listen to Europe's soundtrack music? Simple. Want to play as Poland, but hear Asian peace music and Mesoamerican war music? Simple. You can also make it so that you're listening to the various Leader music throughout the game, which is how I played my last game (and it was really fun getting to hear all the different leader tunes), but that's several degrees more complicated than switching between soundtracks. I'll only talk about the latter here.
Note -- be sure to make backup copies of any and all files you modify.
1. First, navigate to the appropriate XML folder for the civilization you're playing. On my machine (Windows 7 64-bit, using a Steam download), the pathway is: My Computer --> Program Files (x86) --> Steam --> steamapps --> common --> sid meier's civilization V --> assets --> sounds --> XML. (If the civ in question is from one of the expansions or DLC packs, the path is ... assets --> DLC --> Expansion [or Expansion2, or DLC_01, etc.] --> sounds --> XML.)
2. Open the Audio2DScripts.xml file for the civ you're playing. For instance, if you're playing the Aztecs, you'd open AztecAudio2DScripts.xml.
3. You will see two main sections, AZTEC Peace Music and AZTEC War Music. (Continuing the Aztec example.) The sections are populated by <Script2DSound> entries, with one entry for each song played in the soundtrack. For the Aztecs, you will see that there are 47 songs in the Peace section and 47 songs in the War section. What you want to do is change the <SoundID> information for each song so that it matches the geographic region of the soundtrack you wish to listen to.
For instance, you'll see (for the Aztecs) that "SONG_AZTEC_PEACE_1" is tied to the SoundID "SND_AMERICAREGIONS_PEACE_1". All you need to do is change this, so that the song is tied to the "PEACE_1" song from, say, the Europe region. You don't need to copy any music files, move anything around, anything like that; you simply need to rename these SoundID references.
So. Let's say you want to listen to the Europe soundtrack while playing as the Aztecs. Just go down each of the Peace songs and change "SND_AMERICAREGIONS_PEACE_X" to "SND_EUROPE_PEACE_X", where X equals the song number. Then do the same for the War songs: change "SND_AMERICAREGIONS_WAR_X" to "SND_EUROPE_WAR_X". Also do the same for the "EITHER" tracks. Simple, you're finished.
Or, let's say you want to listen to the Europe peacetime soundtrack and the Asia wartime soundtrack. Same idea: change the peacetime SoundIDs to "EUROPE" and the wartime SoundIDs to "ASIA".
It is possible that you will end up with more SoundIDs than there are actual songs in the (new) soundtrack. The AmericaRegions playlist features 24 Peace songs, 24 War songs, and 23 "Either" songs, but maybe the Europe soundtrack has only 22, 21, and 23. (Just a random guess -- I don't have the numbers off the top of my head.) This is not a problem. If the program attempts to call a dummy track, it will simply move on to the next randomly generated song, and nothing bad will occur.
It's possible to use this method to listen to all the soundtracks while playing as one civ. This requires a bit more work, but it's the same principle. Continuing the Aztec example, you'd leave the current lists of Script2DSounds alone, since they will give you the America tracks, and simply add new entries to the end of the lists, via copying and pasting old entries. Just increment the song number by 1 each time and replace AMERICAREGIONS with each of the geographic areas in turn: SoundIDs 1-47 would use the original AMERICAREGIONS (so no changes there), then 48-95 would use EUROPE, etc. Remember to do this for both the Peace and War sections. Again, I'm not sure offhand how many songs each soundtrack contains, but it's easy to find out by peeking into other civs' .xml files.
That's all there is to it. It's a little tedious, doing all the copying and pasting and number incrementing for each of the SoundIDs, and of course you have to do it all over again for each civ you play (if you want the same effect), but it's not hard or tricky or anything. Again, be sure you have backed up the original .xml file(s) before doing this.
By the way, the four different geographic regions are named as follows:
AmericaRegions
Europe
MiddleEastAfrica
Asia
EDIT -- added a missing folder in the pathway for expansion/DLC civs.