Found this problem that mirrors mine exactly: http://serverfault.com/questions/125395/solved-network-issue-ubuntu-8-04-in-vmware-esx
However, VMware tools didn't fix this for me.
Pinging the box from another system on the network gives "Request timeout for icmp_seq xxx". Pinging from the Ubuntu 8.04 box itself yields the same result: no answered pings. DHCP works. The IP shows up in vSphere. VMware Tools is "OK".
Doing an ifdown eth0 followed by ifup eth0 appears to go through the motions and grab a DHCP lease.
arp -a takes a long time to run. Sometimes there are duplicate entries.
VMware tools have been installed & reinstalled.
Interesting: if I ping a different IP each time, I get 2 or 3 responses... and then ping stops responding. Actually, as I watch it... this one got 3 responses, then waited 30 seconds, then 3 more responses, 30 seconds later... 3 more responses. It's like clockwork.