Is there a travel-booking service (like Cheaptickets or Travelocity)
that has a "flexible-destination" tool? I'm given to understand that
in-Europe flights can be very cheap, and I want to go to Rome in
November. So I'd like to find the cheapest flight possible from LA, to
pretty much any point in Europe, from which I can then fly to Rome.
I've already seen lastminute.com, but this isn't really what I'm
looking for, since it's still about six weeks from my intended
departure date.

Howdy,
Within Europe, there are some very cheap flights offered by no-frill
airlines such as Ryanair.  BUT, one usually has to order them in
advance, online, and the number of seats at the cheapest price is
limited, and airport taxes can double some of these prices.  Also,
Ryanair does not fly to major airports, so there is a time and cost
factor that is not so obvious where you are.
The no-frillers have created a lower priced market by major airlines,
but, again, finding a cheap seat when and where you want is risky.

Last minute flights are almost always related to a vacation travel
package including hotel.  The very attractive prices are offered by
travel bureaus that want to fill vacancies in blocks of seats and
rooms that have been paid for, so any paying customer is better than
none.

Personally, I would avoid the hassle and uncertainty and lost time. 
Even with a booked flight, say from Frankfurt to Rome, there would be
a couple of hours wait for the connection and probably two more before
you had your bags in Rome  - and that with jet lag, nine hours time
difference, so if you arrive in the morning after a long flight, you
will be doing this when your body says sleep, with no adrenalin surge
to keep you going.  If you arrive in Rome in the AM, find your hotel
and take a shower, you will be eager to see the sights.
(And if you are doing this with Mrs SBL, ... well, then you'd be
saving two persons the hassle.)
This is just one man's opinion, but I have some experience here in
Europe and with flights to and from CA.
Oh, and in Rome do as the Romans do.  Watch out for your valuables. 
There is a good reason why the inner pockets on Italian mens' jackets
have buttons and zippers.
THis is just a free comment, not an "answer" to your question.
If I can be of further help, please let me know, Myoarin

So I guess the gist of your advice is to skip the hassle and get a
direct flight to Rome because it's worth the extra couple hundred
(which we might not even be able to save anyway)?

and by direct, I mean one ticket, not a nonstop flight.

Yup, trying to be tactful about suggesting how other folks spend their
hard earned money. :)
Every now and then, my wife still likes to remind me about my choices
for cheaper flights to California  - that 3 hr layover in the Twin
Cities 20 years ago with the kids, and other incidents that I prefer
to forget.  ;-/
Yes, I would recommend getting a direct flight, or even a change of
flights in the East if you don't have kids along.  Breaking the flight
in the first half is less of a strain, even relaxing (?), whereas
having a change of flights after 9 hrs and jet lag, with a layover
that is longer than the last bit from London, Frankfurt or Paris to
Rome is tedious  - or worse if anyone just mouths the suggestion:  "we
could be there already ..."
But, again, I am talking about your money.  But since I am, if this is
a two or three week vacation, I would personally consider cutting it a
day short for the expense of a more direct flight.  The trip home via
some less direct route  - if it can be arranged -  would definitely be
tedious, and you may have spent that day in getting to Rome and back
by that route.

Enough.  I hope someone else will comment in order that you get a
broader opinion, and if someone can tell you this is all wrong and
give you information about how a cheap flight can be booked, all the
better for you!
 Good luck, Myoarin

Well, I liked your advice and appreciate it -- not to mention that I
laughed a couple of times reading it, which makes it more than
worthwhile. My wife was of the same mind, and a bit more looking
suggested that there weren't significant savings to be found by the
original method anyway. We had considered taking the trip in stages --
a flight to Dublin or London perhaps, followed by a few days in the
UK, and then on to Rome. But it doesn't look like it would be any
cheaper that way, and for many reasons we're primarily interested in
Rome.

So we booked a straight flight to Rome today with a connection in
Philadelphia. I'm sure we'll be thanking you again when we arrive in
Rome after 17 hours of travel.

Thanks for coming back.  I hope you have a great trip.  I (we :) were
there just a a year ago, but just a hop down from Frankfurt.  I am
sure you will enjoy Rome. Since it seems that you will have more time
there than originally considered, you can widen your touristing
circle, maybe get out of the city for a couple of days to relax and
then return with the nice feeling that you kind of know your way
around and can take in some more Roman history or churches and
museums, or whatever your thing is.

Have a grand time!  Myoarin