I would look for a way to do it with chains, here is why COST!
First, I would look for a way to make chains/cables work. At frist glance on my door, It couldnt be done. When I looked again, there were tabs in place with holes already drilled which allowd me to connect the hooks.
Now for the cost part, I purchased the length of chain I needed for both sides from Walmart in galvenized chain for less than $15, and the hooks I needed from Harbor freight for $4.00. So for less than $20 total, I had the set up.
You could purchase the cable, and the hardware (including the turnbuckle), but that all adds to the cost of the system. Chain is convenient because if one side is just a bit longer than the other, all you have to do is turn the chain one or two turns before you hook it up to even out the length!!!