Cleaning the engine

What is a good way to clean built up grime dust and dirt in the engine bay without causing any damage?

w.o damage make sure your engine is cold and pressure wash it or wash it however you want just make sure you do it in like the morning or something then also make sure your electrical is covered

Cover up all your electricals a anything that could be damaged by water. Spray the whole engine down, really get the dirt and grime out. Then cover the entire thing with armorall and let it sit overnight. Some people are anti armorall but it's like easy off in an oven.
The next morning spray it all off and turn it on. Let it warm up halfway to fully warm, this will dry the water.
And you're done!

Well i've done it dozens of times without covering any electricals. The fuse box and ecu boxes are already covered and the air mass sensor is resistant to water.

Please follow the following procedure (this will have to be done twice if you've not cleaned your engine in a long time, as is clearly the case here):

1. If the engine is hot, best to let it cool down for about 1 hour. Open up the bonnet to speed this along. This is not to avoid damage to the engine, this is to avoid the heat evaporating the engine degreaser too quickly (see #2).
2. Get a can of engine degreaser (its worth the small investment) and spray it everywhere where you can see dirt, and just everywhere, into all nooks and crannies. Spray liberally, use the entire can (do not use two that's overkill).
3. Either :
a.Wait ten minutes, then hose down with cold water. Use lots and lots of water. High pressure is not necessary but medium pressure can be used safely.
OR
b. Get a pail of water and detergent soap (normal detergent will do) and pour it over everything. Then use a sponge, cloth or scrubbing brush to get at dirt in hard to reach places that are visibly dirty. Force may sometimes be needed to get the dirt out, but i suggest you don't use much force at first. Then hose everything down with low-medium pressure water.
4. There's no need to do anything to dry the engine bay fast to avoid damaging something, so there's no need to go for a drive right away. You can leave it wet overnight. No biggie.
5. Wait a day or two, then look at the engine under good light and note where the stubborn stains are. Get them with more engine degreaser, soap and harder scrubbing this time if the surface can take it. If, after a second scrubbing, the dirt does not go away, its become fairly permanent and not to be worried about any further.
6. Do not bother to overly clean the afm. If it looks old and dirty even after hosing it down, leave it that way.
7. Make sure the plastic valve cover (the one with the words 'bmw' on it) has no cracks or breakages. Water getting in there will make its way into the ignition coils and spark plug sockets, causing misfires and poor engine performance. If this happens, no problem. You'll just have to remove the ignition coils and dry out each spark plug socket. Its a pain in the ass but nothing further. Don' ask me how I know this.

That's all there is to it. Do post another picture of your cleaned up engine bay once you're done.

Thanks guys...now it's about finding sometime to try these methods out. J have so much work to do on this car...maybe tomorrow morning since it's a holiday.. Thanks again folks!!!

On second thought, in order not to waste the engine degreaser, why don't you do an ordinary water hosedown right tonight? It would get much of the dirt off. By the time you come back to it tomorrow, it would be dry and so wouldn't dilute the degreaser which would be a waste.

Degreasers are great! I heard also to avoid spraying it on any hoses.