You have what we in the networking business call a "black hole". Basically you can ping from the 2.x network to 1.x because the 2.x computer asks the belkin router (the computers gateway to the outside world), the belkin router might know, but is probably going to try and arp to find the 1.x computer address. The computer responds to the belkin router, which then forwards back to the 2.x computer.
The other direction the 1.x computer asks the linksys router where it is and (unless you have set up a route to 192.168.2.0/24 network if the router supports it) the linksys router has no idea. Now a 192.168.2.x address is a private class C (which can't be routed accross the internet) So the linksys router might just give up then, or it might try to send it up the chain (the linksys router asking ITs gatweway) at some point they are going to stop trying to resolve the private address and return back destination unreachable.
Perhaps you installed the belkin router solely as a way to isolate the wireless clients, not a bad practivce, You're either going to need to add the route from the linksys router to the 2.x network (may not even be supported on home devices without custom firmware) you would also need to make sure that NAT is disabled on the belkin router.
To have the networks talking directly (not seperate networks) , you are going to need to do a few things. Change the way that the belkin is connected. You're probably in the WAN/Uplink/Trunk/Modem ports on the belkin, change that to on of the regular/LAN/Downstream ports. You would also need to turn off the DHCP server (or configure it as a relay) and change your address on the belkin to the same range as the linksys (I'd suggested belkin is 1.2 since linksys is 1.1) This woulld allow traffic to still be resolved through the linksys router, DHCP is handled by linksys, belkin may be a wireless AP, but DHCP should still be through linksys for the wireless clients. Basically we changed belkin from being a SUB network to an Access point and switch on the same network.