The engine itself is not the only thing that can cause a noise under your hood. There are also accessories driven by the engine that can make noise, such as the AC compressor, the power steering pump, the alternator, the water pump, or any other belt-driven device. These can be difficult to track down, because the noise often travels through the drive belt; in fact, it could be the drive belt itself that's making the noise.
To find a noise like this, you can do a couple of quick tests. Say you suspect that the AC compressor is making the noise. With the engine running, turn the AC on and off. If the noise goes away when you turn the AC off, then you know it likely has something to do with the AC compressor or something in the AC system.
With the power steering pump, try turning the steering wheel from lock to lock. If the noise gets louder or changes when you do this, it could mean a problem with the power steering.
As for the alternator, water pump, auto tensioner, or other belt-driven accessories, you need to put your ears on it. There are a couple of different ways I do this. The first is with an automotive stethoscope. You can often find these at your local auto parts store, and they aren't that expensive.
These can be used to find all kinds of noises, not just the ones in your engine compartment. Another method that also works is to use a long screwdriver, the longer the better. You place the tip of the screwdriver against the part you suspect is making the noise, then place your ear on the screwdriver handle. This amplifies the noise and makes it easier to hear.
The purpose of both of these tools is the same: pinpointing the noise. With a little detective work, you can often find the source of a noise. Once you find the source, you can begin to address it. Here's a video about drive belts that offers more information on finding noises in the engine-driven accessories.
One other method is to remove the drive belts one at a time, and run the engine after you remove each belt. Of course, this only works if you have multiple drive belts. This method can help you pin down a problem accessory or belt. Don't worry; the engine can run for short periods of time without these accessories just fine. Just don't do it too long; run the engine just long enough to determine whether or not you've found the noise.
If you have a timing cover over your water pump and timing belt tensioners, this can make finding these noises more difficult. In cases like this, I sometimes remove part or all of the timing cover and run the engine. This can help you pinpoint the source of the noise. This might be more trouble than it's worth, depending on the situation. If the timing cover is difficult to remove, you might just stick with an educated guess and prepare yourself to replace the timing belt, water pump, and its tensioners to address the noise.
One last thing while we're on the subject of drive belts and engine noises. I have sometimes seen the harmonic balancer or crank pulley break and cause a noise. These often have a layer of rubber between the inner and outer part of the pulley. Sometimes this rubber breaks or deteriorates, and as a result causes all kinds of noises.
If you hear a noise on the front of the engine, it's a good idea to check for this problem. It's usually much easier to do after you remove the drive belts.