Do you really want a deciduous hedge if it's for privacy? If you do, I can suggest a few things in my yard that the deer have left alone:

Hibiscus syriacus - nice flowers, and they'll grow faster than burning bush Euonymus
Spiraea 'Antony Waterer' - again nice flowers, and the deer absolutely leave them alone
Peking Cotoneaster - more of a typical hedge shrub, with fast growth and nice shape. I have found some minor deer nibbles on them, but not much
Golden Vicary Privet - no deer damage at all and grow to 5' pretty quickly
Red twig dogwood or yellow twig dogwood - both great shrubs that can make a fine hedge, and they both have nice looking red branches in the winter

To tell you the truth, though, I haven't noticed deer eating my burning bushes.

Thanks for the suggestions! What can you suggest that is not a deciduous hedge? If you can give me some ideas I would be very pleased.
I was glad to hear you say that you have not had a problem with the deer and your burning bushes. I have one out front and so do many around the area and they look fine. ?? I was reading on line that the deer love burning bushes. Perhaps deer in other states who know.

The evergreen hedge is a tougher one to figure out. We have some great evergreen laurels at my house, but the deer LOVE those things! My laurels are growing pretty well now, but that's mostly because our deer population has shrunk in recent years with more houses being built all around and up the hill from us. The deer are mostly here in the winter, but for some reason, I only get a few in my backyard these days, where the laurels are.

I know some people make hedges out of arborvitae, but I read that some varieties are attractive to deer. You might do a little research and find that some kinds are OK. I've been wanting to plant a few Thuja 'Green Giant' trees, which are a kind of arborvitae, and I've read that some people cut them into hedges. They are said to be deer resistant, but I think that's not the same as deer proof. Depends on how desperate the deer are.

I have some leatherleaf viburnums (Viburnum rhytidophylloides) that are almost evergreen, and the deer have left them alone. They have large leaves that hang on the plant in the winter and give more privacy than some other deciduous shrubs. They are evergreen in milder climates. They grow fast, too.

Someone gave me a book on deer-proofing the garden, but it was after I already bought all my landscape plants, so I never used it. Can't say how good it is, but you might find a good book at Amazon.com if you search for deer proof plants or deer proof gardens. Good luck!

I found this post for red, deer-resistant plants. There are some listed that you may be able to make a hedge out of.

http://forums2.gardenweb.com/forums/load/peren/msg111346154000.html?21

Thanks so much to both of you for you help. I do appreciate all the info you gave me and I will read further and find just the bush that would work. I looked at some of the ones that you mentioned that were not evergreens and they sure were pretty. I have enjoyed looking at your landscape stevation and it is certainly beautiful.