"I need to insulate under my raised home in New Orleans. My initial choice was spray foam insulation, but after speaking to plumbers, electricians and termite treaters, I've ruled that out. This leaves regular bat insulation or styrofoam insulation sheets. I'm really leaning towards the styrofoam sheets. The decision now is to cut each sheet to fit between the joists or to lay the sheets over the joists. Any opinions?"

If the electrician & plumber do their jobs corretly there
shouldn't be issues with foam. Granted you would serve yourself & them well by taking pictures..electrical, plumbing & (crossing my fingers that you will say that your
ductwork & heating system are not under your home!!) ductwork, supply boxes & plenums.

I know in a perfect world..but still make sure that these guys are licensed, bonded & insured...lots of fly by night
contractors preying on our hurricane destroyed areas.

That said...
I have both options & opinions..
First understand that what keeps batt insulation from performing to it's optimun is the air flow across or through it. Air robs insulation of its insulating value.
So keeping it in a sealed area...such as with in the joist bay with a foam sheathing board nailed to the underside of the joist is the only way to go that route.
I can't remember how high y'all have to raise your homes,
but around here 3' to 4' is the height. (and you get $$ off your homeowners insurance if you raise it higher)

So assuming your home is raised enough for a workable height this is what I would do. First take pics before you start any work..then once each step is completed. These are for future refrence.
Go under your home & seal all the holes you see that enter into your living space. Plumbers, bless their hearts, cut a 4"x4" hole for a 2" pipe, and the holes cut for bath drains...don't get me started.
Seal large holes with metal flashing caulk around the opening & further out..depending on how large your flashing piece needs to be. This will be your air seal.
Screw the flashing in place & caulk around the pipe where it penetrates the flashing.
Some holes will require less drastic sealing & for these I recommend caulk. Caulk will expand & contract with the movement of the building materials. Great stuff will overfill these holes, and after a few years the great stuff will break down & pull away from the wood.
If the gap/hole is larger than 1/4" it is too large for caulk alone. This is where backer rods come in use. Backer rod is a round tube of foam sort of like pipe insulation.
It comes in a few sizes 1/4" & 1/2" is what I use. If the gap is larger than this....use the flashing.

If you find it too hard to use flashing...use some foam board & button cap nails to close the larger holes.
And caulk it.
Use a caulk with a long life..don't want to do this again
& some areas will not be this acessable again.

Once these areas are sealed. Then you insulate.

Use an unfaced batt insulation..the unfaced can be ordered if not in stock, but there is a whole issue about vapor barriers that make the unfaced a must.
Cut the insulation & fit it well around all the pipes & wires.
Depending on the size of your floor joists upsize the insulation. If your joists are 2x6 don't use R-19..use
R-25. If the insulation does not keep in contact with the
floor condensation will occur & cause a world of trouble.
There are friction fit insulation batts..but hard to find
but the store where you purchase your unfaced batts will have wires to fit between joists to hold the batts in place.

Once you have an area of insulation installed then come with the foam boards. Use the termite resistant ones...
(and Thanks for the link MightyAnvil...I'll be buying some of this myself!! Great info, I didn't know it existed!!)
I'd stay with the dow product & get tongue & groove
and install this under the joists. I tape the seams with a
tyvek tape..or similar product. The surfaces have to be clean & dry or tape will not adhere,
And I caulk the pieces that are around the exterior of the home.
This may be overkill...but I like to have secondary lines of defense. Mainly I want to stop ALL air infiltration that will keep the insulation from performing.
sierraeast's idea of washers & screws is a good one.
I would do these @ areas where plumbing would need to be accessed..under sinks, toilets, tub/showers & washing machine.

I would also make it clear to anyone that they are to leave the foam barrier intact once they finish with it.
You make a hole...it becomes your hole...fix it.
(you may want to have the termite foam board info for them
or even keep some for them to use)

Foam insulation...well I'm not a big fan of it..mostly based on install issues, overselling & sales bs..but I do recommend it quite often as an under floor install.
I've never had problems with electricians plumbers are anyone who knew their trade. Usually lack of dealing with a 'new' product is what causes people to not be happy to have to deal with it.
But 2" under your floor would take care of the air sealing of most of the holes.
More than that & gravity will take its toll within a few years.
You would still have to catch some of the larger areas like under tubs. And if your home is ballon framed (wall cavaties open from crawlspace to attic
pretty common in many New Orleans homes) these areas would need to be sealed ...again...foam board & button cap nails would be sufficient the spray foam over this would close the cracks. I've even used window screen for balloon walls
cut, staple in place & let them foam away.

Whichever route you choose, take the time to document, seal
document again & insulate.

Best of luck to you.
And if you are on Decatur...visit Central Gro & have one for me!!