This write-up is kind of from another forum but I used a different method when detailing the engine bay.
Micro Fiber Cloths
Paper Towels if you don't have Micro Fiber Cloths
Garden Hose with water
Before you start the entire process if you do not have a cleaner/degreaser you can run to your local Dollar Store to pick up some Awesome which works FANTASTIC.
To start off with you will need to tape/bag off things in your engine bay that does not take kindly to water, for example the battery, alternator, air filter (if running open cone), PCM and all the other good stuff.
Next you will want to get your degreaser (Awesome) and spray it onto the part/area you want to clean.
After you spray the cleaner on their you will want to use a paint brush (doesn't matter on size) to agitate the soil/dirt on the surface.
Once agitated you will take the garden hose and lightly spray it onto what you have agitated.
After you spray it off with the garden hose you can wipe it off with a microfiber cloth.
Repeat the above steps until it is totally clean.
Enjoy a nice clean engine bay.
Here are some before and after pictures
And here you can tell the difference between where it was cleaned and agitated versus nothing.