The CV34s are pretty easy to disassemble. Do one side at a time so that if you get stuck somewhere, you have a reference to put them back together. Also, make sure you download the service manual from the articles link above. Also, as F64 noted, you don't have to separate the carbs. In fact, avoid this if you can. Otherwise you will need to re-sync them.
I've taken mine apart probably 20 or 30 times, so I'm pretty well familiar with their internals, and if you have issues, I can help you out.
I know you mentioned that you went through the ignition system with F64. Did you check that the advance unit was free and returning to the correct neutral position? If it's stuck advanced, then the bike will not want to run below 3000-3400 RPM very well. The cam should turn freely and smoothly. I wrap the cam in rubber and use pliers to rotate it. Be careful not to mar the surface; I'm not sure but there might be easier ways to rotate an advance from a bike with electronic ignition; my '80 has points, but your '81 should have electronic. The big point is not to damage anything.
It might also be a good idea to replace your Carburetor holders (the rubber boots which hold the carbs to the engine). If those are leaking, then the resulting vacuum leak will make the bike run poorly/not at all at low RPM. Z1 usually has them available. They're a bit pricey, but worth if for the sealing. When you get them, be careful not to overtighten the clamps that hold the carbs on; the holders can split if you do. Tighten the clamps until they're just snug; they only keep the carbs from backing out, they don't add any additional sealing.