Edit
Try this solution from here
Install the game. (if already installed please backup your saved games, uninstall the game, delete the files remaining in the folder that was installed and install the game again)
Download the update 1.09 from Blizzard Support.
Once downloaded run the update as administrator. (remember that the game disc must be in the drive)
Download this file DiabloFIX.zip.
Extract and copy the file to the game folder.
Edit your game shortcut
In Games Explorer: First open Games Explorer and wait until the game update the stats, now close and go to C:\Users\username\AppData\Local\Microsoft\Windows\GameExplorer sort the folders by date and it should be the one down the GamesBoxArt folder, open the folder {#-#-#-#-#}>PlayTasks>0 right-click in the game icon and go to properties, replace the .exe for .bat at the end of the target path
or
In Start Menu: Go to All Programs>Diablo> right-click in the game icon and go to properties, replace the .exe for .bat at the end of the target path
Make sure that the game is not setting to run in compatibility mode or disabling settings.
My original answer was for Diablo 2. My mistake. Sorry.
Edit
Also I had to disable desktop composition and run the game in Windows XP compatibility mode. HTH.
This happened to me and my solution was to use the 3Dfx Glide Library on Windows 7. This link was able to help me solve my issues.