I've seen several reports lately of the newest Radeon graphic drivers not working correctly with SimCity. The "fix" that seems to work is to install an older driver to use with SimCity. Since it was working well with your older card, I would try using the same version driver you were using with that one.
If it works, let us know which driver it was. The last driver reported to fix it was non-whql-win7-64bit-radeon-software-crimson-relive-17.6.2-june13, that's obviously really old. If there's one that's newer that works, it would be good to have a confirmed newer driver to suggest in the future.
(I know that installing an older driver is not an optimal solution. It's just that SimCity is an older game and EA is no longer issuing updates to it, so it will not be something they will fix. You could try seeing if AMD will do something to support SimCity and issue a fix for their newest cards and drivers.)