Thanks for all the replies guys, I now know that the outlets are not
grounded. I borrowed a friend's device which determines this and used
it.

I really can't do anything which involves running a wire to a pipe. My
landlord lives right behind me and has warned me that he will not
accept anything unless it is approved and done by his electrician.

Andy you mentioned that there are UPS packages that can provide
protection. Could you point me to one of these? I haven't had any
luck finding any.

I would say that we get maybe 3-4 storms a year...not sure if thats
considered a lot as I've never lived outside hawaii. We have tiled
floors so I guess building up static while walking is less likely? I
can tell you that within the last year (at my old appartment which is
only a few blocks from this one), I've had at least one occurence of
the power blinking out or dimming...which I'm guessing are signs of a
surge.

Now I know that many of you have replied with suggestions and have
answered my question, I appreciate your time very much and I really
want to hear from as many sources as I can get before making this
decision.

I guess I could pose the question this way. If you had $2000 worth of
computer equipment and got 3-4 storms a year, would you risk running it
on ungrounded outlets? Or would you rather move?

Thanks again,
Jim