It depends on whether the forces involved are "conservative" or "dissipative".
Conservative forces are forces that you can describe with a potential energy function - for example, the gravitational force, or the interaction of two electrically charged objects.
They are called conservative because energy is conserved when you are moving in the potential field. So, if you go back to the configuration you started in, the energy is exactly the same as when you started. This is why they say work is path-independent in a conservative field - it's the same thing.
On the other hand, there are lots of processes that are non-conservative. Anything that dissipates energy in a random way is non-conservative. An example is stirring a glass of water. You use some energy to spin your spoon around in the water, which drives the water around in random ways.
The thing you are pushing against is the viscosity of the water, and if you stop stirring and put your spoon back to the place you've started, energy has been lost! In fact, the energy's gone to heating up the water. So there you've done some work in a cyclic process.
It turns out that almost any real process is dissipative! There is almost always some friction - air resistance, electrical resistance, sliding friction, viscosity.. You name it, if it involves the motion of something, some fraction of your energy is going to be dissipated away and turned into heat. In fact, it is very difficult to eliminate these losses, despite the best efforts of many engineers.
This is what leads to the statement of the second law of thermodynamics - in any process, the entropy of the universe either remains constant or (almost always) increases. Entropy is a measure of the disorder in some system, and when you dissipate some energy, you do so in a random, disordered way! So by stirring your spoon you contribute to the increase of entropy in the universe.
You can sort of look at entropy and energy as being different types of the same sort of thing - energy is what it starts out as, and entropy is what it becomes. But it's a one-way street. Once you turn energy into entropy, you can't turn it back. People have speculated that this fundamental feature of our universe is what makes time seem to go in only one direction. Sorry, I gues I'm starting to veer off into philosophy now.. But hopefully this helps you understand what "work" means, which is just a way of saying "energy" while specifying that it's mechanical energy.
Anyway, the point is that almost any physical process will have some inefficiency or loss associated with it. This loss is a loss of energy, so I would submit that actually NO cyclic processes do zero work, but some do in an approximate "physics class" sort of sense, where you neglect frictional losses.