The LMS 3.1 data sheet does not mention win2k8 at all, so it's safe to assume "no".
However LMS 3.2 is supported on select 2k8 servers:
Your LMS 3.1 license(s) can be used with 3.2, the latter of which is downloadable at: http://www.cisco.com/go/lms
yjdabear is right. You need LMS 3.2 for Windows 2008 support. LMS 3.1 will not work.
Even with LMS 3.2, there are two caveats. One is that you have to manually configure swap or Daemon Manager will not start. This is done under the advanced settings in the System control panel. Your maximum allowed swap should be inline with the LMS requires.
The other caveat doesn't affect everyone, but unfortunately, the true cause is as yet unknown. We have a workaround, though. If you find you cannot login to LMS on 2008, you may be hit by CSCtb17186:
After entering credentials into the login screen of CiscoWorks LAN Management Solution, an error is thrown indicated that redirection has failed. Alternatively, the user may be redirected forever. Firefox may report:
The page isn't redirecting properly
This is most readily seen with LMS 3.2 running on Windows 2008 servers. However, the problem can affect any version of LMS running on Windows if cookies are disabled for the internal SYSTEM user on the LMS server. One known way to disable cookies for this user is by using the Internet Explorer Admin KIT (IEAK).
First, make sure the server is running a supported configuration. Right now, that means IE 6.0 or 7.0. If LMS is installed on Windows 2008, make sure the server is running SP1 and NOT SP2.
If the problem persists, go the Services control panel, and get properties for the CiscoWorks Webserver service. Click the Logon tab. Change the login user to be Administrator, and enter the proper Administrator password. This MUST be the LOCAL Administrator account. Then, login as Administrator, and verify that cookies are allowed in the Internet Options control panel. Then restart CiscoWorks Daemon Manager.