Do any languages have clear morpho-syntactic constructions that mark backgrounding and foregrounding?

If I understand the question correctly, then I'd say all languages have some way of doing this. (Let's not forget intonation.) In English, you can say "It's John, I'm talking about." Instead of "Today I'm talking about John." In Czech, you simply change the SVO to OVS word order.

But I think the way you phrase the question has to do more with information structure (e.g. topic-focus) of discourse. Foregrounding/backgrounding also happens at phrasal/lecical level. E.g. I'm standing in front of a Church. vs. The church is behind me. vs. The're a bench in front of the church. vs. There's a Church behind this bench. You'll find this sort of analysis in cognitive/construction grammar approaches.

I'd be surprised if there were no languages that did some of this purely morphologically but I can't think of any examples, right now.