The car makes a distinctive rattling type of sound which seems to be coming from the front and the top of the car, from the engine area. The sound is intermittent and I usually do not hear it. When I drive over bumps and rough pavement, that seems to bring it on. I opened the hood and the front section which holds two fans and the opening for radiator fluid is loose. I could move it somewhat back and forth, but it did not make the sound when I did so.
If you can get it to make this noise when idling in the driveway (may have to rev the engine a little), you can then use a length of old garden hose as a sort of stethoscope to narrow down the cause. If it only occurs when moving you’ll have to get someone experienced (like a shop tech) to ride along with you. Could be a problem in the suspension – safety issue – so good idea to not defer on this.
If it’s a metallic rattle, it could be a loose heat shield (or a small rock that got between the heat shield and the converter somehow - like happened to me).
Previously, I drove another Civic, and a mechanic on that one told me that the heat shield was loose, and so he removed it. I took the present car to a mechanic, who drove it, but the noise was not present. He said the radiator is mounted on rubber, so it is normal for it to be slightly loose. He looked over the engine and put the car up on the rack, but found nothing loose. I am not going to pay for new struts even if they have wear, because I don’t think they are the problem and because I am not going to have the car forever. The noise seems to be somewhere on top, but it is hard to tell. The noise mostly happens when I first start out and am driving slowly, or if I drive over rough pavement and bumps, which obviously shake up the vehicle. Thank you for the above suggestions.
Bad suspension components can rattle over bumps. Check the shock mounts. Stabilizer links can do this too if they’re loose. Tie rod ends as well. Basically anything loose under there, so get it up on jack stands and shake stuff until you find what wiggles.
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I think you nailed it with “basically anything loose under there, so shake stuff until you find what wiggles.” 
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I presume you mean the radiator (and its fans) isn’t solidly mounted in place. It’s true that the radiator is on rubber mounts, but it shouldn’t feel so loose you can move it about. Is this an aftermarket radiator? When I installed an aftermarket radiator on my Corolla I had to add some home-brew spacers where it is held at the top to prevent the radiator from being too loose and rattling.