Usually, rather than triangulating algebraic varieties, it is better to find [cylindrical decompositions][1]. These are finite cellular decompositions that exist for every real semi-algebraic set and thus, restricting scalars, for every complex variety. See [Algorithms in Real Algebraic Geometry][2] for more.

If you want a proof of finiteness of $H^{\ast}_{DR}$ without resolution of singularities and without topological tools, you can see Monsky, [Finiteness of de Rham Cohomology][3]. I wouldn't call it simple, though!


  [1]: http://mathworld.wolfram.com/CylindricalAlgebraicDecomposition.html
  [2]: http://www.amazon.com/Algorithms-Algebraic-Geometry-Computation-Mathematics/dp/3540330984
  [3]: http://www.ams.org/mathscinet-getitem?mr=0301017%20.