Sharif,
Normally, ground plane on one side gives most of the shielding benefits, and leaves the other side free for tracking. You can put extra bits of ground copper between the tracks on the wiring side, but it makes almost no difference, as they are beside the signal tracks, not above or below them, and so they only block a negigible angle.
You really need to go to multi-layer boards if you want shielding both sides, so you can use the inner layers for wiring; this is not really practical for DIY construction, so one ground plane is a good compromise.
In practice, both my old hand made prototype, and the production Maclaren units were very quite when built with single ground planes..