Here is my situation:

We have several long-range (several kms) networks that are connected
via 802.11b wireless.  I have done this with Debian based gateways at
each site.  I have successfully setup the routing so that GRE tunnels
exist between all networks.  My problem is that I would rather all
networks belong to one large subnet (like a university).

Currently networks exist on various subnets ie:

Network A is 10.0.10.0/24, Network B is 192.168.22.0/24, Network C is
192.168.56.0/24 etc.  There is no problem with routing at the moment. 
I can ping a host in Network A from Network C.

As this is a community non-profit project, people require games.  One
of my problems is that various games will only search their local
subnet for existing game sessions.  I need to put everyone on a large
subnet.  I was thinking of readdressing ALL networks to something like
172.16.0.0/16 and somehow using bridging to get networks to talk.  For
example each site will only contain a 172.16.16.0/24 or 172.16.32.0/24
worth of systems, but their netmask will specify the whole class B.  I
understand some sort of proxy arp and bridging solution will allow me
to make the networks appear as if they are all one physical network?

My idea is something like this:

Network A:   Hosts: 172.16.16.1 - 176.16.16.254 / GW 172.16.16.253 /
NM 255.255.0.0
Network B:   Hosts: 172.16.32.1 - 172.16.32.254 / GW 172.16.32.253 /
NM 255.255.0.0
Network C:   Hosts: 172.16.48.1 - 172.16.32.254 / GW 172.16.48.253 /
NM 255.255.0.0

Another nice feature would be to let users browse all SMB based
systems with the Network Neighbourhood, be it via a Wins server or
Netbios broadcasts.

All networks would host their own local DHCP, local squid and a
central DNS would be used for name resolution.

I just need a yes or no and the software / method that will be
required.  Alternatively a better way to accomplish this.   Not an
overly detailed answer!  Following that I have no problem
reconfiguring 2 networks to test if this will work.  Preferably linux
to be used for everything (can't afford Cisco gear for community based
stuff!).  I have no problem implementing any GRE or IPSEC tunnels if
this is required.

Yes, this is possible, using the 802.1d bridging feature of the Linux
kernel and the "brctl" command. You may need to enable the bridging
option if it is not compiled into your kernel or available as
the "bridge" module.