I have the Deluxe version (Sim City 4 + Rush Hour) and the below code is from the Graphics Rules.sgr file. I had the same problem(s) as others in this thread and initially tried to correct it according to "shadowdanes" instructions. I settled on a different solution that did not involve sacrificing shadows.
Works great now.
PS! I have marked the changes I made in the code with the color red (Almost at the bottom).
EDIT: Seems there is a limit to how long a post can be in this forum. I've added an url to a copy of the file if someone is interested and only added a short snippet below of the code that I changed. Download file from here.
# The Radeon series has a very slow path for partial
# depth buffer copies. This has apparently been
# fixed for 9500 and later models in the latest drivers,
# but cannot be fixed on previous Radeons because of
# hardware limitations.
# NOTE: disabling this rule unnecessarily may lead to
# very poor graphics performance.
#partialRule "Radeon"
#stringMatch cardIdentity "*Radeon*"
partialRule "SlowSubrectDepthCopy" -any
stringMatch cardIdentity "*7?00*"
stringMatch cardIdentity "*8?00*"
stringMatch cardIdentity "*9000*"
stringMatch cardIdentity "*9100*"
stringMatch cardIdentity "*9200*"
stringMatch cardIdentity "*9300*"
stringMatch cardIdentity "*9400*"
# stringMatch cardIdentity "ATI, Radeon"
stringMatch cardIdentity "ATI, Radeon VE"
stringMatch cardIdentity "ATI, Radeon Mobility"
stringMatch cardIdentity "ATI, Radeon FireGL"
# results in special low-impact scrolling mode.
property noPartialBackingStoreCopies true
# try to reduce the number of dirty rects.
property dirtyRectMergeFrames 8
# use black and white cursors. The Radeons seem
# to work with colour for a while, but in low
# memory situations revert to flickering.
option $Cursor $CursorBW
#end
end