Pointed to a tree in the woods by next door neighbor, it is a Mulberry, probably bird propagated, Can cuttings be rooted?

yes, but it's not easy. i killed all mine this yr. i have few things to try next yr.

Mulberry can be rooted but with lower success. Don't expect it to root as easily as fig cuttings.

I had one out of 4 that took root, but it never grew any more this summer.  Leaves recently fell off from cooler weather.  Not sure if it will come to life next year.

Anyone try rooting in water?

If you try the baggie method, allow for more moisture than for figs. My bags had a ton of condensation inside, which seemed to work out well for mulberries and pomegranates. The mulberry cuttings will develop roots and immature berries. Just pick off the fruits and wipe off any mold. I vented them about once per week. The cuttings will also develop big, crusty calluses, but this can look like a fungus. Don't freak out. It's normal and good.

trying to root Mull's here too and out of 4 none are rooted (in nursery soil).
tried in water but they rot....

i am growing them just for curiosity like the lemon guava's (red and green from seed).

Also be sure its productive there are mullberry trees that are non fruiting (male?)

Out of about 60 I tried to root and 10 that I tried to air-layer, I have one to show for my effort.  I think the key to success is timing.  Mulberries seem to fruit early.  If you try to propagate at the same time as figs, the cutting tries to fruit and it seemingly runs out of energy.  My guess (and it is only a guess at this point) is the best time to try to root is after fruiting in late spring.

I have limited luck with some cultivar. The Shangri La, and Middleton I think I rooted 3of5 each with scoring and hormone straight into potting mix. I can't guess how many Illinois Everbearing and
Pakistan I have tried both dormant and summer green cuttings without a plant to show. Last year I tried several cultivar in sphagnum moss just like fig, they callused up like crazy and rooted but only a single Kokuso No. 20  survived into a plant for me. I will probably try the increased moisture as noted above, but not sure at the point of losing potted up rooted cuttings it will matter. I have failed 100% on several other cultivar. They graft easy for me I just need to find some wild seedlings to graft to. They are pests in some areas, but I don't have any around me.

  I rooted mulberry cuttings in a vase of water on Northside of the house behind an  AC unit,   I cant remember how many cuttings I began with but 5 grew.

It all depends on the type of mulberry.  Some root very easily and other varieties are very difficult.  

Here are images from the mullberries I did this year. Strip the entire way around the bark about an inch or two down to the hard wood core (past the "green" cambium layer), rub the wound with honey to sterilize, fit a water bottle around and filled bottle with damp peat moss, taped bottle with clear packing tape. Only took a month and the bottle was full of roots. Potted up and the leaves will drop but then the buds will swell and new leaves will pop out. I feel like this method is better than cuttings since I did not have to do anything after the initial set up and got nice big trees.