RollerCoaster Tycoon 1&2 require graphic card drivers version 13.4 or older.

Try using a DirectDraw wrapper.
http://bitpatch.com/ddwrapper.html

Extract the included ddraw.dll and aqrit.cfg files into the game's directory (where the RCT.exe / RCT2.exe is)

Open aqrit.cfg with a text editor and set
ColorFix = 1
Save it and run the game.

If it doesn't work, also set
ForceDirectDrawEmulation = 1
Save and try again.

Also try the newest Catalyst drivers as they fixed some DirectX 9 issues.
(32-bit version / 64-bit version)

Since I updated my drivers I also have this issue... but thanks for the suggestion with DirectDraw wrapper, will try it out!

Doesn't work for me, unfortunately.

Try updating DirectX.
Windows 7 doesn't ship with DirectX 9, so maybe that is it.

Then it's definitely worth checking an older driver version, even if I like this option least. There are known issues with newer AMD cards, recent drivers and older DirectX games.

I don't think an installation of DirectX 9 will help in this case, but it also never hurts. As Strijkbout pointed out, Windows 7 and newer are missing some older libraries by default.

Since I play a lot of old games on my computer, I have of course DirectX 9 installed. I only have this problem with the newest graphics drivers. But I'm not that eager to play RCT2 at the moment, it's just something I noticed. Once I want to play it again I will first try the XP mode of Windows 7 and if that doesn't work I can always play it on my laptop.

edit: it works in XP mode, with a few drawbacks, like apparently non-scaleable resolution. I used to play in 1280x720 on my Full HD screen which worked really good, but now the game gets centered in the screen with the maximum resolution of 1280x1024, which leaves black borders all around and makes everything really small. Well, I hope there will be a solution besides going back to old drivers in the future.

Seems to work again with the newest drivers (14.4).

Nice, thanks for the heads-up.