samba - Jul 2009 - Samba Windows resolve issue

I am having an issue with a new samba setup.  My gateway server (H1 - the
new setup) is running Samba and acting as my networks wins server.  It also
houses my email and web server.  It is connected to a private network to
another server (H3) running Samba acting as my main file server.

The issue that I am experiencing is that H3 can view our webpage that is
hosted on H1 but my Windows machines that are connected to H3's subnet can
not.  If I try to ping H1 from H3 it resolves the name properly.  If I ping
H1 from the windows boxes it returns the external IP address and times out
on the requests.  i hope that I have presented that clearly.

H1 is fully qualified in the host files on H1 and H3.

H1's smb.conf

 [global]
        workgroup = admin
        server string = Web Server
        netbios name = h1
        netbios aliases = mailserver
        interfaces = lo 192.168.4.0/24
        hosts allow = 127. 192.168.1. 192.168.4. 192.168.5. 192.168.6.
        log file = /var/log/samba/%m.log
        max log size = 50
        encrypt passwords = yes
        security = user
        passdb backend = smbpasswd
        smbpasswd file = /etc/samba/smbpasswd
        domain master = no
        local master = no
        os level = 32
        preferred master = no
        wins support = yes
        name resolve order = hosts wins lmhosts bcast
        wins proxy = yes
        dns proxy = yes

H3's smb.conf

[global]
   netbios name = h3
   workgroup = ADMIN
   server string = Administration Samba Server
   security = user
   hosts allow = 192.168.4. 192.168.5. 192.168.6. 192.168.1. 192.168.0. 127.
   log file = /var/log/samba/%m.log
   max log size = 50
   passdb backend = smbpasswd
   encrypt passwords = yes
   smb passwd file = /etc/samba/smbpasswd
   interfaces = 192.168.4.3 192.168.6.1 127.0.0.1
   remote announce = 192.168.5.255 192.168.4.255 192.168.0.255
   local master = yes
   domain master = yes
   domain logons = yes
   logon path    logon home    logon script = %U.bat
   wins server = 192.168.4.1
   wins proxy = yes
   dns proxy = yes
  add machine script = /usr/sbin/adduser -n -g machines -c Machine -d
/dev/null -s /bin/false %u

I left out the Samba shares but I think any main configurations are present.

This machine is a rebuild and I didn't have this issue before.  The main
difference is that this server now has two external facing nics to two
different interent connections.  I don't think routing is the issue because
H3 can resolve H1's name correctly and display the web page.
Thanks for any help anyone can give me!

Does anybody have a clue what the issue might be?  Maybe even a hint as to
what I could look into?  Is there more information that I could supply that
would help?

I am stumped and my endusers are frustrated.

Thanks again - Doug