I have a GeForce FX 5200 Ultra, so, this shouldn't be too terribly different than what you have to do...
First, go into RivaTuner.
Go to the Direct3D tweaks.
Next, go to the Textures tab.
Under Texture format settings, uncheck "Enabled 8-bit palettized textures"
Click okay, restart your computer if it asks you to.
Then, open up the FF7 Config.
Select 640x480 Full Screen.
Check the NVIDIA box, and mark the TNT option.
That should take away the gridlines for you. It did for me.