First, the Uniform Commercial Code statute of limitations as enacted in your state applies; a check is generally not treated as a contract at all, except by metaphor. The standard rule is [used to be four years, now six], although TX may have picked a different SOL without having told me about it. But that's a civil matter - what it means is that they can file suit to recover the money at any time, subject to your right to defend on the ground of statute of limitations if the period has expired as of the date they filed suit. It's an affirmative defense and doesn't apply unless you make it do so by filing a written pleading with the court. No judge anywhere will say, "Oh, plaintiff, you can't do that, the SOL has expired."
As to the criminal matter, it depends on whether the amount of the check places it in the misdemeanor or felony category - there are usually two different statutes of limitations for those. In Virginia, for example, in cases of misdemeanors (i.e., the check was for less than $200), the SOL is one year.
If the check was post-dated, then it was a loan, not a check, for purposes of criminal prosecution.
In order to sue on a bad check in a civil case, the plaintiff has to jump through certain hoops, called presentment and notice of dishonor. In theory, they have to present the maker with the check and demand the payment. In Va. that's done by a demand via registered mail, without any real presentment; but you are entitled to the check back if you pay it. I don't know what the statutory hoops are where you live, but I'd bet there are some, and they don't have a right to sue on the check unless and until they comply with the law.
If the person who is attempting to collect is not the party to whom the check was made out, they may qualify as a "debt collector" under the Fair Debt Collection Practices Act. If it were me, I would assume that they do qualify and send them a demand for verification letter. You can find out lots more about the fdcpa in this website.