The idea of having a solid ground plane is to provide the least amount of impedance on the ground routing. But stitching together single local ground areas by some vias is a violation of this idea, because vias heavily increase the impedance again. A ground plane does only work if it is really solid! Even if you only make some interruptions and gaps then the effect is a huge suffer in performance.
Have a look at

http://www.8052.com/forum/read.phtml?id=103336

to see what a solid ground plane is for.
If you "must" use a double sided board, then the best way to get the circuit working, is to use a solid ground plane on the bottom side (especially when using SMDs), and running all the signals on the top side (component side). If signals must cross each other, then use bridges (wires) on the bottom side placed over the solid ground plane, without interrupting it! Keep the ground plane at the bottom side always as solid as ever possible!
Each ground pin of SMDs is connected to ground plane on bottom side by at least two vias placed directly at according ground pin.
This methode only works with simple boards, where the pinout of chips allows a bus wise routing of address and data buses and if only chips are used which aren't faster than 74HCMOS logic. Keep also in mind, that doing the artwork for such a construction can be more expensive than using a multilayerboard from the start. (This is only true if artworker's salary is higher than manufacturing costs of multilayer.)

Kai