If you checked no cables and wires are loose, then it's not the case. The case is just a box keeping the components in place, and it's not really connected to the functional parts of the computer aside from power and reset buttons and leds. CPU is hard to test individually unless you have another compatible computer, and even then it would be a pain in the ass with all the CPU cooler removing and reinstalling. But since you already replaced the mobo, it means if everything else works, then the CPU is the culprit. PSU you could also test with another computer and much more easily than a CPU. It's also possible to test the PSU rudimentarily totally outside of the computer. You'd just need to short-circuit two specific wires of the big mobo connector, which is a signal to the PSU that the computer is turned on. After that you could test there's current available in the various cables with a multimeter, spare fans, and whatever runs with the available voltages.

PSU problems are annoying in the sense than when a PSU dies, it can take a bunch of other things with it. The cheaper the PSU, the higher the danger.

Other than that, you could try to remove all removable components one at a time and see if it makes any difference. Unless you already did that.