In order to allow computers in the 192.168.0.x LAN to ping computers in the 192.168.1.x LAN, routerB must be configured to forward ICMP traffic between its WAN and LAN.
To do this, go to portforward.com, and search for instructions by routerB's model number. they have instructions for most common router models. configure routerB's NAT to allow ICMP echo packets, and you should then be able to ping inbound to routerB's LAN.
To understand whats happening, you must first understand that your computers are not on the same LAN, so in order to ping each other, routerB must pass the traffic between them. Your router however, sounds like a standard home/small-office router, which implements a pair of technologies called Stateful Packet Inspection, and Network Address Translation.
Every router has at least two networks, its inside (LAN), and its outside (WAN; everything outside the LAN, from its perspective). from routerA's perspective, its WAN is your ISPs network and the Internet at large, and its LAN is everything in the 192.168.0.x/24 network. routerB's LAN is the 192.168.1.x/24 network, and its WAN is routerA's LAN, the ISP netowrk, and everything else outside its LAN. So computers in routerA's LAN are in routerB's WAN.
NAT and SPI generally allow all hosts inside the LAN to connect to computers on the outside, and receive responses to those connections, but it prevents a computer on the WAN from initiating a connection to a computer on the LAN. This is why you can ping from 192.168.1.x (routerB's LAN) outward to 192.168.0.x(routerA's LAN and routerB's WAN). It is also why a computer in the routerB's WAN cannot connect inbound to a computer in routerB's LAN. You can however create a forwarding rule in NAT that allows specific types of connections to pass through the SPI, and initiate connections on the inside.
see more info here: http://en.wikipedia.org/wiki/Port_forwarding