First of all, when Seiberg's papers say that a non-renormalization theorem may be proved by holomorphy, be sure that Seiberg is right. But it doesn't mean that holomorphy is the only argument that may become crucial in a proof of such theorems.
For example, a complex spinor may decompose into real or pseudoreal spinors in a lower dimension but some of the non-renormalization theorems may still apply. Also, the BPS states (annihilated by a subset of supercharges) have masses that are totally dictated by the "central charges" and don't receive any quantum corrections; this argument is independent of holomorphy. It simply follows from the vanishing of some $QQ$ bilinears in the state, which - by SUSY algebra - may be rewritten as a difference between the energy/mass and some central charges.
Other proofs of non-renormalization theorems may involve perturbative arguments that things cancel at each order because SUSY prohibits the "wrong terms" in each case, because of some dimensional analysis. In principle, those arguments - although they are familiar in the holomorphic context - don't depend on holomorphy, either.
I think it is misleading to ask about all non-renormalization theorems simultaneously and expect that there is a single word, such as holomorphy, that contains proofs to all of them. Different non-renormalization theorems have different proofs that use different ideas: the required ideas depend not only on the dimension but also on the precise SUSY algebra, precise theory, and even the precise quantity whose non-renormalization we're proving. And in each case, there may exist several highly inequivalent proofs of the same proposition. You would have to ask more specific questions to get more specific answers.