In this case you would probably want to use a 24-bit subnet mask (255.255.255.0) Your router will need to have a logical interface for each network. So it would have an interface address on the 192.168.1.x, 192.168.2.x and 192.168.3.x network. Once you have that created on the router, the devices on the LAN would point to the logical interface address as their default gateway.
It sounds like you do have a good plan, however, you are restricted to only 252 additional devices in this configuration since the .0 address would be the network and the .255 is broadcast and the .1 address is the router interface in each subnet. You could consider using a 20 bit subnet mask and then you would have 4094 hosts per subnet. A sample subnet would be 192.168.0.1- 192.168.15.254