Well there are a few things that you might run into trouble with using other speakers with a z-5500 setup.
-sensitivity mismatch
-impedence mismatch
-crossover mismatch
For sensitivity mismatch, all you'd have to do to compensate is in your soundcard controls, adjust the output level of the FR/FL so that they'll match the volume output of the center channel given the same signal. You'd want to do this anyway with the speakers given that the distance from the listening position will probably be different for each speakers (especially the surrounds) unless you have an ideal setup where you were able to get them the right distance on stands already.
For impedance mismatch, giving the z-5500s a 4-ohm speaker to drive rather than a 8-ohm speaker would make the amp work harder to drive those speakers. I haven't tried to run a 4-ohm speaker off a logitech setup so I'm not sure how well they'd handle that. A 4-ohm speaker will be a more difficult load for the Logitech set to drive, but you could certainly try it. The amp on the sub might get hotter, you might run into distortion quicker on the front two speakers, and you might end up blowing the z-5500 fuse. If you keep volumes reasonable and aren't trying to set any records with them, then you should be able to run it with those speakers though.
For the crossover mismatch, I'm not sure what the crossover specs are on the Edifier setup, but I'm guessing it's going to be pretty comparable. If there were a big crossover difference, you could either be not getting the full benefit of the speaker change, or you might be asking the speakers to play frequencies lower than they're designed to do so which could lead to some issues. I think you're probably safe on this point though.