<p>How would you go about creating a random alpha-numeric string that matches a certain regular expression?</p>

<p>This is specifically for creating initial passwords that fulfill regular password requirements.</p>

If you have a specific problem, you probably have a specific regular expression in mind. I would take that regular expression, work out what it means in simple human terms, and work from there.
I suspect it's
possible
to create a general regex random match generator, but it's likely to be
much
more work than just handling a specific case - even if that case changes a few times a year.
(Actually, it may not be possible to generate random matches in the most general sense - I have a vague memory that the problem of "does any string match this regex" is the halting problem in disguise. With a very cut-down regex language you may have more luck though.)