I\'d like to program a .NET app that provides a 3D render of an animated world. I\'m coding on a Windows PC, but many of my target audience will be on Linux.
There
Maybe not quite what you're looking for but, Unity3D is a cross platform 3D gaming environment which is based on Mono, which means that you can code the bits you need to in C#. There are indie and pro versions with the indie version being free. http://unity3d.com/
Use OpenTK
I'd second the idea to look at Unity, which if I remember rightly uses C# as a scripting language to let you write the game code (using Mono), allowing the actual rendering engine and so on to use native code (presumably C++).
Otherwise, I'd be wary of using Mono for anything like 3D graphics or other 'hard core' uses... anything involving GUI or rendering I'd be nervous. But that's my personal choice, if you've time to spend researching it might be worth throwing a prototype together... but i would not write a lot of C# code without the ability to test it on Linux very early in case there are big issues.
You might look into using Qt and OpenGL. I imagine, however, that OpenGL is probably too low-level (read: "needlessly complex") for your particular needs.