Hello all! I have only recently started playing Oblivion again. I have been having some trouble with random crashes and cannot play a full hour without a crash. My computer is most likely not the problem as I can run games like Bad Company 2, Dawn of War 2, Shogun Total War.

Any advice as to how I could remedy this?

Try posting your Load Order, or a DX dialogue.
You could reduce your screen resolution, play with the sliders for how close an object has to be before it becomes visible in game.
If you have Oblivion Mod Manager use the view Load Order in Import/Export, If you have Wrye Bash use the list mods function to get a current Load Order to copy and paste.