Cannot connect to VMWare virtual machine through NAT

I've run into similar issues when setting up domains. By default Windows 7 is set to reject PING requests as well as other requests. Depending on your Linux distribution it may be set to reject as well. (but doesn't seen to be in this case if I'm reading your post correctly)

If you really want to see a response from Windows you need to take down the firewall (Not advised but for the purpose of testing this is fine). Go into Windows Firewall with Advanced Security, then turn off the corresponding firewall for your network (or all of them if you can't figure it out).

As another option, you can create an exception for PINGs in the inbound/outbound rules.

Bridged mode has the VMware client "share" the network connection with the host and it acts as if it is a machine on the same LAN. It requests an IP address and everything. Then communication between the two "machines" goes through your router.
(http://www.vmware.com/support/ws4/doc/network_bridged_ws.html)

By using NAT mode, VMware uses your host PC as a "router" of sorts and assigns it an IP address inside that LAN. So all communication is direct in this case. But it actually does do full translation of IP addresses/everything when attempting to access outside the network (can access internet).
(http://www.vmware.com/support/ws55/doc/ws_net_configurations_nat.html)

Host mode creates a direct connection between the two computers that cannot go beyond that. By default no DHCP is enabled so you need to set that up manually.

Any should work for your purposes but it's important to be aware of the differences in how messages are transferred and IP addresses are assigned.