Ok, here how I managed to get it going.
Get 2 thumb drives. First of 8GB and another one bigger for the actual system.
1. Restore the "InstallESD.dmg" from Mountain Lion to the 8GB flash (as explained many times)
2. Find any mac that is authorized to boot in Mountain Lion and have both flash drives with you. Start that mac with option, select the 8GB drive and go through the process of installing ML onto the bigger flash drive (I suggest 32GB flash, or at least 16GB).
3.When done, boot boot from that bigger flash drive, go though the process of creating an account and so on. Once in the finder, I suggest deactivating as many things as possible not be of weight later in the process (Disable notifications, Wdgets, Autoupdates... all small little things, it is not obligatory, but makes the process faster later on.) Also connect to the wireless to have it ready.
4. Now shut down that borrowed mac you don need it anymore.
5. Plug the bigger flash drive in your Lion computer. Get the zipped file described in this thread. Replace the files in /System/Library/CoreServices and in /usr/standalone/i386
6. Get Kext Wizard and install following files into the big flash drive:
(all are located in System/Library/Extensions in your Lion installation you install them to your bigger Mountain Lion external flash drive in the corresponding place)
AppleIntelGMAX3100.kext
AppleIntelGMAX3100FB.kext
AppleIntelGMAX3100GA.plugin
AppleIntelGMAX3100GLDriver.bundle
AppleIntelGMAX3100VADriver.bundle
Now for the audio I installed the same way also that "hacked" audio driver from the creator of this thread and I placed it also on the Extensions together with the other .kext files, but I have no audio. It shows changes in the volume when I click on the keyboard, but no sound. In the control panel all the different alerts are dimmed out.
NOw you're ready. Restart and boot from the external bigger flash drive in Mountain Lion. Works with the following problems:
* No sound
* No brightness control (the screen is not at max as previously stated, but at a normal level)
* There is a slight graphics glitch in notifications, which is not too bad I guess a bug that no one corrected for this type of machine. Other than this graphics are smooth and accelerated. THey seem to work even better than Lion on the same machine. Mind you I have 4gb of RAM
Now you can make a bootable backup with carbonecompycloner or whatever and you're done so you could instal on your main drive at any given moment.
I would gladly accept any help in fixing audio and video, for the rest the machine works like a charm under Mountain Lion.
Ok upon further investigation I have following points to report.
1. All programs work (make sure you change in the "Security and Privacy" CP that all applications can be installed, so Gate Keeper is disabled
2. Video: kext are not loaded (checked in the sys profiler), so I guess those intel .kext in my guide are not needed so far because they don't load. The display os recognized as internal with 64mb of shared ram, where 3100 Intel chip with loaded drivers is capable of using 144mb of shared memory. So that explains the graph glitches and all, if we manage to load the drivers, video will work like a charm. I'm able to play 1080p movies now, but the refresh rate is not smooth, it lacks video memory.
3. Audio: strange enough in the sys profiler audio is recognized exactly as in Lion, it lets me change volume, but I have no sound.
4. The system reports I'm running a much earlier build of Mountain Lion than what is installed ( I think it was 26A128p) which must be because of the hacked files we installed earlier.
5. That big external flash drive you created can not be put back and booted in the original Mountain Lion capable mac that created it in the first place. Whatever changes we made, we have locked that installation only to a particular group of older hardware. I'm saying this because in Lion and before we were able to boot an external installation in any mac that supported that OS
OK FINAL EDIT
After using the tool and the hacked kext provided by hackerwayne, now I have audio too. The only remaining problem as he indicated is the video not loading the proper drivers because they must be 32bit. The rest loads in 64bit and it works really sweet. I hope for any help on the video drivers and the macbook 3.1 will be one sweet machine under Mountain Lion.
On a side note it is a real shame from apple not to support those models under ML. It would have taken them what an hour of coding. They are becoming very greedy. That's why after being so loyal to apple for over 16 years I bought Samsung Galaxy S3 and let me tell you something I'm never looking back to the iPhone that I keep in the drawer. A new king is in town and Apple better shape up their act. Fame comes and goes fast.
