How do you carry traditional snow shoes, is there a carrying belt or harness or something?

They go on your back.  Which is a real problem if you have a rifle there.  In this case you bring a back pack and strap them to that..... this gets them away from your body and gives some room for the rifle on your shoulder.

If you are passing areas where you want them and areas where you don't want them - pull a small sled.  Even a cheap yellow WalMart one can be very handy so long as you are not passing thick brush.

If you are passing through brush - you can wrap them in a small tarp - tarps pull effortless across snow and if you wrap your stuff so the front is kind pointy it will follow you like a puppy though most all places.

Some of this depends on the model you have.  They make very light weight aluminum and rubber 'rescue' ones to take with you 'just in case'.  These will be easy to pack.  The big long wooden ones with uncured leather I have are a real pain to pack anywhere - awesome when in use - terrible to lug!