Please note that not "any old phone" will work in Europe, even if unlocked. The phone must be not only GSM (T-Mobile or AT&T in the US), it must also have the right frequencies to work in Europe, which doesn't use the same GSM frequencies used in the US. The key frequency you need for voice service is 900MHZ in much of Europe, but old GSM flip phones used in the US may only do 850MHZ (plus a few higher frequencies).
I have two older T-Mobile flip phones, and neither of them worked by default in Europe. I was able to hack the older one to do 900MHZ instead of 850MHZ and used it a few times successfully. I turned on the other one (which could not be hacked) once for fun when I had it with me in Belgium, assuming it wouldn't work, and indeed, it did not pick up anything.
If the old phone is a "quad band" or "world" phone it will probably work. The best way to be sure is to google the exact model number and find its specs. If it has 850MHZ but not 900MHZ, it probably won't work in Europe (not everywhere, anyway).
Most smart phones do seem to have the 900MHZ GSM frequency, though.