Non-local admin access to the network

I have a PDC and 2 BDCs in a decent NT network. I run static IPs, DNS and WINS. I have been upgrading the Windows 95 PCs to XP lately. The only problem I have is that unless the user that logs in to the PC is in the Local Admin group, they can’t access anything through the network. If they try pinging anything, it eventually times out. They can’t browse over the network either. When I add [domain]\domain users to the local admin group, and they log in, they can then access servers and apps over the network. My problem, is that I do not want them to be local admins, the closest thing I would like them to be is a regular user. Is there a way to do this without resorting to 3rd party software?