I can list all the steps here, you might need to use a little bit of trail and error since everything is set up a little different. But this is what works on my setups, at least.
1) Open Internet Explorer. You must be using Internet Explorer (you cannot use another browser). IE8, the version that comes with Windows 7, or IE7 should work fine. You should be using a disk with the NTFS filesystem in order for this to work.
2) From the menu on the right, Tools->Internet Options. This brings up the "Internet Options" dialog. In the dialog that comes up, under "Browsing History", press Settings. For "Disk space to use", set the maximum of 1024. Take a note of the text below "Current Location". It might be something like:
"C:\Documents and Settings\Pig\Local Settings\Temporary Internet Files" if you are using XP. If you are using Windows 7, it will be a little different and longer, but regardless the last directory will typically be "Temporary Internet Files". Just make a note of it, you'll need it later.
3) Back to the "Internet Options" dialog. Press the "Delete" button under "Browsing History". This next dialog works a little different for different versions of IE, but basically you want to use it to delete "Temporary Internet Files". In some cases you'll press the "Delete files" button then confirm, in other cases "Delete Temporary Internet Files" is a box you check, after which you hit the "Delete..." button or some such.
4) Now go ahead and browse to the page. It's important to let all 'streaming' media files download completely, which can take a minute or two. When everything is done downloading, go to the next step. Don't do any other browsing (on Internet Explorer). You can use another browser if you like.
5) Open windows command line. Navigate to the directory noted at the end of step #2. Now you need to navigate to some non-indexed directories. They won't show up with the normal dir command or any other windows calls. Use a dir /ai to find them and navigate down into them. The first one will typically be something like "Low" and the next one should be "Content.IE5".
6) Now that you have navigated into "Content.IE5"l, you need to find the media file. If you do a dir /ai from Content.IE5, you'll see any number of non-indexed directories show up. They all look like random combinations of letters and numbers, fortunately they're only 8 characters long. The name of the media file is set by the site itself, for instance Nico Nico Douga uses smile*, Youtube uses vidplayback* or videoplayback*, etc. But typically the media file is so much larger than all the other files you shouldn't have too much of a problem finding it even if you don't know the name.
Take advantage of dir /s to find these files. For instance, "dir /s | more" will list off all the files in all subdirectories and then you can look for a really big file (typically in the range of 10-100mb) which will be the media file. Once you know the naming convention of the website you are using, you can do something like "dir /s vid*" to show all files with that name in any subdirectory under the one you are currently in.
7) Once you find the media file you must copy it out of there to a directory you can see. Make a directory off your c: root like c:\scratch and copy it there. Once you have this figured out, you can write a batch script to do this for you. Generally your command might look something like:
copy \3j9dvxn6\vid* \scratch
8 ) Once the media file is copied out of IE's cache, you will likely need to give it an extension. Sometimes, in the case of .mp4 files this will not be necessary, but in the case of most .swf, .flv, and other files, they will have no extension when thy are copied out so you will have to add it yourself. As for how you know which extension to add, trial and error is the best I can do. Nico Nico Douga uses mostly .flv but will occasionally use .swf. Youtube uses .mp4 for most HD and .flv for most everything else.
9) Windows Media Player Classic HomeCinema can play back most of these file formats, like .swf.
10) Once you have your media file, you'll want to go back to step #2 and clear out the cache again. If you do this just before getting a new media file, it makes it quicker since you don't have to look through as much junk.
11) You can use other web browsers to do this, there are instruction online to do this that work. I tried using Chrome and Firefox to do his. The problem is that these drivers will not cache large media files, and there doesn't seem to be any way to set the max cacheable file size limit. For Firefox it is around 64mb, for Chrome it is around 40mb. These sizes quickly get exceeded these days, especially when dealing with HD video.