Engine mounts are low on the list here for a reason. Worn or broken engine mounts can cause a rough idle, but they are not the first place I go when diagnosing an idle problem. My method is to exhaust all possibilities with engine performance before I start looking at the engine mounts. Many people come to me and say they've replaced their engine mounts and they still have a rough idle. Engine mounts aren't always easy to replace, so that might be a lot of wasted work on their part. I then suggest much of what is listed above, and they eventually find the cause.
The takeaway here is that worn or broken engine mounts can cause an idle issue, but I would suggest you look there last when dealing with an idle issue. Get your engine running smoothly before you look to the engine mounts.
As I said at the beginning of this article, many things can affect the idle. Start with the obvious: Check the oil and check the computer for any stored codes. From there, look for things that can offset the air fuel mixture, such as vacuum leaks or perhaps leaking injectors. Don't forget about the engine’s mechanical function; if the engine isn't healthy, it likely won't idle well.
In addition to the items mentioned above, I've written two articles specifically on Honda idle problems: