In his book Complex Geometry, Huybrechts states and proves a classical theorem of Siegel that a compact complex manifold of complex dimension n has algebraic dimension at most n. Unfortunately, I don't understand his proof, so I'd like another reference - but it's proving surprisingly difficult to turn up. I can't find it in Griffiths-Harris or Voisin. Did I just miss it?
Try Shafarevich's Basic Algebraic Geometry II: Schemes and Complex Manifolds. The result you're looking for is Theorem 3 on page 175.