I just tested the following, but I think will you need to upgrade to Version 17.
You can add multiple zones in JRiver by right clicking playing now and selecting Add Zone. Add six zones. To keep track during setup, you can name them Soundcard1 Left, Soundcard1 Right, Soundcard2 Left, etc.
Set the volume for each zone to Internal Volume and turn the master volume control up all the way. This will allow you to control the volume of each zone independently.
Set the audio output for each zone by going to Tools > Options > Audio Output and set the output mode to Direct Sound. In the Output Mode Settings set the soundcard for that zone.
For each zone set the Output Format to 2 Channels using Player > DSP Studio > Output Mode
Now you need to make the left and right channels be able to play different stuff. To do this go to Player > DSP Studio > Parametric EQ. For Soundcard1 Left you would add Mix Channels and add the right speaker to the left speaker. Then add Mute and mute the right speaker. For Soundcard1 Right you add the left speaker to the right speaker and then mute the left speaker. This will combine the stereo signal into a mono signal and make each zone only play to one output.
You can go back and rename your zones to something more helpful like Living Room, Kitchen, etc.
Now you can play to each zone separately and control the volume for each zone. You can use your computer, laptop, Android phone, iPhone, or other device to control each zone. If you want to play the same thing in multiple zones you can link the zones. You can right click a linked zone and select Adjust Link Timing in order to make sure the zones are playing exactly synchronized.