unfortunately, you are going to have to do a BUNCH of tests to make sure that there is no issue with your system (and I don't know how much gear you have) - before you may come to the realization that the RCA jacks on the K3 box have snapped internally, and you are not getting any audio output from them. This is not as easy a test, as with a Kona LHe or LHi, unfortunately.
I urge you to try this, if you have an SDI or HD-SDI source (like a VTR). Close FCP, open the control panel, and play a SDI or HD-SDI tape into the Kona 3 with embedded audio - do you hear any audio ? Do you have the ability to observe an AES/EBU audio source (like a HD or SDI VTR with AES input ?) - play tone out of FCP, and see if the AES/EBU output works (and shows up on your VTR meters). You did not specify your exact equipment that you own, so it is tough to guess what you have to test with. Perhaps you have very little equipment to do tests with.
In FCP, you should certainly trash your FCP preferences, trash your AJA Kona preferences, even uninstall the Kona drivers, and reinstall them.
I too have gone thru this (but I had more gear, like an HD VTR to do a bunch of tests with). Ultimately, it wound up being bad RCA jacks on the K3 box, and I had to get AJA to replace the box. But you need to go thru all these steps, because you will be frustrated if they send you a new K3 box, and you still have no audio.
All my cables are connected, speakers on, volume up, etc. So I'll have to go through this long process you outline. I actually expected as much, just thought I'd throw it out there in case this was a known issue.
I have the LHi. Some times my audio stops working, too. It's usually fixed by relaunching FCP or rebooting. Just a bug, I suppose. It comes and goes, and I haven't noticed a pattern yet.
I also have a Tapco LinkUSB audio interface attached. So, I can just change my audio prefs in FCP to use the LinkUSB and bypass the Kona, if relaunching or rebooting don't fix it.
MacPro (Harpertown-Early 2008) 2x4 3GHz; 32G RAM all the same brand; 10.5.8; QT 7.6.4; FCS3; Kona LHi in PCI slot 3; Primary display: 30" ACD; Secondary: HP LP2480zx DreamColor via HDMI to LHi and DVI to MacPro.