I recently got Pomegranate cuttings and just wondering if I could use the Spaghnam Moss method for them like fig cuttings. 

I have yes I scored them and used dip-n-growand into the moss. They callused up, and some started roots but I lost several at cup stage. The later batch I let callus up in the moss then stuck direct in 1 gals and I lost less. many folks have rooted them straight into pot with no prep and good results. I am holding off and doing mine closer to warm weather.

Is it okay if they root horizontally? I do it with fig cuttings

I did the vertical, but last year I did my figs vertical as well. I use Gal baggies with the moss and top of bag open in a storage Bin, and keep everything vertical, different cultivars by bag. open the bin with bags already open easy to let fresh air in. I got less rooting along the whole fig cutting than when I had horizontal.  I have only done Pomegaranate vertical.

They will root in mist sand and using most of the same methods that we use for figs. I have heard that bottom heat will greatly improve rooting in pomegranates and mulberries.

I will root it vertically in the bag method. They may be too long to fit in 1 gallon bag, if that is the case, I will root in a longer container.

To me, they are easier to root than figs. Just stick in potting soil or in ground will work. Or use fig methods.

They root easily in potting soil and perlite.  EASY!

I've done several hundred poms but they don't root as easy for me as figs.  I've given up using rooting hormone on figs this year after last year's problems but am stilling using it for poms.  I'm not doing as many poms this year but am having good results just sticking them in pots with ProMix HP and wrapping the tops with Parafilm (eliminates most of the need for misting).  I have a thread showing how I've done that with figs and the poms are right next to them on heat mats set at 85F.  Last year I had one of my poms really grow fast and was nearly 6' by the end of the year.  Never had that kind of results with any other poms before.

I have tried to root my pom as soft wood cuttings in early summer with no success. So, is now the right time, then?

Even though they don't seem to grow roots nearly as fast, I seem to have better luck with pomegranate cuttings than I do with fig cuttings.

Dormant cuttings now, pencil size or a bit larger.  Use IBA of about 4500 ppm, bottom heat.  Soft-seeded varieties root easier than hard-seeded varieties.

Jon, regarding mulberries.  Morus negra are hard to root and bottom heat and IBA will probably help but Morus alba 'Oscar' certainly grows like a weed and roots very easily and I've heard other albas are all easy.  A few years ago I had some extra cuttings laying around in the shade in the spring for maybe a month and then decided to just push them in the ground where they were. Both grew so I had to dig them up and pot them up since they were under a 100 year old pear tree.

stick'em directly to the ground in a warm sheltered area and forget about it.

I have some POM cuttings ordered from UCDavis. I hope they get here soon! I think I only get three of each variety.