If we had a singular verb here the difference between the sentences might become clearer.
Periodically check with Microsoft online for a newer version of this software.
Periodically check with Microsoft online for the newer version of this software.
Now the question is whether we ought to be using the indefinite article (a), or the definite article (the).
Whatever newer version you may find, it has not been released yet. In fact it may not even exist yet -- and even if it did, we don't know about it. Therefore we cannot talk about it in definite terms.
If I was looking at your computer and noticing it was running slowly, I may say "You need a newer version of this software." If I was aware that a newer version existed (regardless of whether it's been released or only announced), and thought you needed that specifically, only then would I say "You need the newer version of this software."
In the original example we are talking about hypothetical, future software versions which the user must check for. This is a case for the indefinite article, not the definite.
The only further complication comes in when we return to our plural noun. There is no plural indefinite article, rather the article is just omitted, leaving us with the sentence you started with.
Periodically check with Microsoft online for newer versions of this software.
This sentence is correct as is.