The links to the tool in that article are dead, and the Easy Transfer doesn't really offer an option to transfer just the profiles.
This tool seems to be similar to the "WSR Profile Tool" mentioned, judging by its description it may do what you want, provided the old machine is still functioning. Though it's listed as working for XP.
EDIT: The tool errored out on my machine, but it pointed to the files at "%LOCALAPPDATA%\Microsoft\Speech\Files", which also stores the audio from speech training. I don't know whether transferring those would work by itself.
EDIT#2: As I expected, there's a reference to those files in the registry.
The key on my machine is "HKEY_CURRENT_USER\Software\Microsoft\Speech\RecoProfiles\Tokens\{1119241B-E581-422C-B9B9-1038E81E8FBD}\{DAC9F469-0C67-4643-9258-87EC128C5941}\Files", and contains a list of names prefixed with "AM" and suffixed ".am", "am_bak", ".tpc", ".fpr", or ".fpr_bak", referencing the files in the "Microsoft\Speech\Files\MSASR" folder in the data field.
That key is partially unique; "{1119241B-E581-422C-B9B9-1038E81E8FBD}" is the ID on my machine(it could be the ID of the speech profile itself), yours will be different, but if you navigate to "HKEY_CURRENT_USER\Software\Microsoft\Speech\RecoProfiles\Tokens\" there should only be one key anyway, so it's not hard to find
({DAC9F469-0C67-4643-9258-87EC128C5941} is not unique, as it's the same on my Windows 7 VM).
The same key is also found under "HKEY_USERS", but that's because "HKEY_CURRENT_USER" is a reference pointing to the currently active user's ID(you can copy or modify either, they're the exact same data).
There's also a similar key that references the training audio files under "HKEY_CURRENT_USER\Software\Microsoft\Speech\RecoProfiles\Tokens\{1119241B-E581-422C-B9B9-1038E81E8FBD}\{00000000-0000-0000-0000-000000000000}\Files" that has key prefixed with "TrainingAudio-" pointing to the "TrainingAudio" directories' files(as you'd expect).
Again, "{1119241B-E581-422C-B9B9-1038E81E8FBD}" is unique to my machine.
So in summary: If you export/import the registry keys under "HKEY_CURRENT_USER\Software\Microsoft\Speech\RecoProfiles\Tokens" and copy/paste the files in "%LOCALAPPDATA%\Microsoft\Speech\Files", you could potentially have a working copy of your speech profile on another machine.
As a side note, there are also references to the "AM" prefixed name under "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Speech\Recognizers\Tokens\MS-1033-80-DESK\Models\1033\L1033\AMs", which in turn point to files in "%windir%\Speech\Engines\SR\en-US\", however those also exist on a Windows 7 VM I have(which doesn't actually have a trained speech engine), so there shouldn't be a need to copy those files or that key.