Yes, drop ceilings usually need about 3" of room to work. You may have the need to pop the tiles out, and 3" is cutting it close. If a tile gets punctured, you can 'break' it up in place but you need to get a new one in there.
So, what are your options? I opt to install drywall. And where there's a shut-off for, say, a water line, I either move it to another location or I put in a false vent. You're probably saying, "What if there's a leak?" Yes, the drywall will get ruined, and you'll have to cut out the bad drywall and replace it and re-prime/paint it. With a drop ceiling, even if you had more room, you'll have to replace the tiles. ALL of them because over time, they fade. In other words, if say 10 tiles get ruined, it's easy to replace them. But, you'll have a contrast difference between the replacement tiles and the existing ones around them. See what I mean?
What you can do is re-run any wiring so that it all runs together and not all over the place. The same for the plumbing. The end results will be all your electrical and plumbing hardware will be 'localized' between, say, 2 sets of joists. As for the ductwork, if it hands below the joists, you can box it in. That's easy. Other than that, not much else to say unless you have more questions.
My best to ya and hope this helps.
PS: Don't use 3/8" drywall on the ceiling. It will sag on you over time.