Cannot ping hostname.  Hostname not resolving. (unknown host)

My server is running Linux, but most of the network is Windows, including the DHCP and DNS servers.

The Linux box get its IP via DHCP.  I'm pretty sure this works (it gets xxx.xxx.105.24).

Here are results of various ping tests:
                             |
ping to Linux box IP:  
     - Linux box to itself - YES
     - Any external box to Linux box (regardless of subnet) - YES

ping using Linux box hostname:
     - Linux box to itself - NO !
     - Windows boxes on same subnet (xxx.xxx.105.xxx) - YES!
     - Linux boxes on same subnet - NO !
     - Boxes on a different subnet - NO

I'm sending the hostname to the DHCP server.  Here is what the /etc/dhclient-eth0.conf file looks like:
send host-name "name";

Here is what /var/lib/dhcp/dhclient-eth0.leases  file looks like:

lease {
  interface "eth0";
  fixed-address xxx.xxx.105.24;
  option subnet-mask 255.255.255.0;
  option routers xxx.xxx.105.254;
  option dhcp-lease-time 604800;
  option dhcp-message-type 5;
  option domain-name-servers xxx.xxx.109.126,xxx.xxx.109.38,xxx.xxx.26.41;
  option dhcp-server-identifier xxx.xxx.109.126;
  option dhcp-renewal-time 302400;
  option ntp-servers xxx.xxx.252.4;
  option dhcp-rebinding-time 529200;
  option domain-name "companyname.com";
  renew 0 2005/12/4 17:52:48;
  rebind 3 2005/12/7 20:28:39;
  expire 4 2005/12/8 17:28:39;

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