Hi,
As a starting point, make sure first of all that your LinksMMIII exe file is set to run with Administrator privilege.
Make sure you have patched up to 1.07 to address various issues and to give you access to changing game graphic resolutions easily through control rather than via registry edit.
Then, have a look with game (Graphics set-up) at the various resolutions available and, where possible, match that to your native desktop resolution. Most graphic drivers shouldn't pose a problem, but Links in its raw state does need help to cope with the newer cards, so the patching is required.
Also look at the top of this forum area and that of the
main board to patch for Links issues such as viewing conditions crash and server updates.
That is the best starting point to work from, so that we rule out a few of the more routine issues. Please do post back to tell of progress or lack of and the 'Brains Trust' here at LSPN will, I'm sure, be able to help out with some of the deeper issues you may encounter.
EDIT: Using the same card and driver set-up on Windows 7 environment here, so know I know it should be sound. Hopefully having kit that matches will mean we can get things working fairly quickly.