Are you using forms-based authentication? If so, make sure you have Service Pack 2 loaded, as it made some big changes related to Office Client integration. From TechNet:
http://technet.microsoft.com/en-us/library/cc262201.aspx
If you have not installed Microsoft Office SharePoint Server 2007 with Service Pack 2 (SP2), client integration is disabled by default when you use forms-based authentication. This is because client integration does not natively support forms-based authentication prior to Office SharePoint Server 2007 with SP2. When client integration is disabled, links to client applications are not visible and documents cannot be opened in client applications; documents can only be opened in a Web browser. However, users can download documents, edit them in client applications locally, and then upload them to the site.
If you have installed Office SharePoint Server 2007 with SP2, client integration client integration is supported for Word, Excel, PowerPoint, and SharePoint Designer authoring.
Actually, now that I think about it, the answer might actually be even easier. What happens if you navigate to the SharePoint drafts folder, open the document, and then check it in directly from within Word (as opposed to checking it in from the SharePoint UI)?
First off, you need to make sure that you have an MX record somewhere in DNS so that a mail server can deliver to the sharepoint server. This could be a public DNS record, or internal, depending on your needs.
To test whether you have set up Sharepoint and the local SMTP server properly try the following:
View the folder you configured the smtp server to deliver to (C:\Inetpub\mailroot\Drop for example) - you will watch this to see if messages are delivered.
Then open a telnet session to the sharepoint server on port 25. Via this you can "send" a test message directly to the smtp server using the address of your Document Library.
telnet spserver.localdomain.local 25
ehlo test.com
MAIL FROM: <test@test.com>
RCPT TO: <doclib@spserver.localdomain.local>
DATA
Subject: Test Message
This is a test message
.
If you have your smtp server configured correctly, and you see a message saying the message was queued after typing the above, then you should see a file appear in the folder you are watching.
If Sharepoint is configured to check this folder correctly, after a few minutes you should see the message disappear from the folder and appear in the Document Library.
The only time I have seen messages disappear from the drop folder, and not appear in the Doc Library, it has been when sending messages with nothing in the body. Sharepoint seems to discard these, even if there is an attachment.
I also had issues where Sharepoint was creating contacts in AD with the wrong email address - check these are created as you would expect otherwise Exchange may discard mail coming in from outside.
Best Answer
In the root of the document library select Actions -> "Open in Windows Explorer" and drag and drop or cut and paste as you normally would.
Sorry, just saw this was 2010. It's under the Library tab. Open with Explorer.