We are going to be in Prague, Vienna, Budapest and Salzburg among other places on our upcoming trip. We are vegetarians and would appreciate any suggestions.
Thanks

Convince yourself that sausages aren't meat! Get prepared for large amounts of cabbage.

At least while in Austria go to the farmer markets in the early AM. Some outside the city. But there you can find lots of fresh vegetables. But as far as restarants. I being one who eats little meat. Asked for salads and Bread and avoided most meats. They had great vegetable soups in most places too.

Check out this site: http://www.happycow.net/
It has lists of veg. restaurants in most cities, with reviews. I'm vegetarian (though not vegan) and just returned from visiting the places you mention and didn't have much of a problem, though when out of the cities a picnic is probably the best way to go. There is lots of great fresh fruit, veg, cheese, breads etc at the markets. Restaurants offering traditional menus were pretty much out of the question, but there are lots of other choices -salads, sandwiches, pasta, pizza, falafel etc. - in all of these cities. While in Budapest, don't miss Vegetarium at 5, Cukor Utca 5 - outstanding.

The Best Western zum Hirschen in Salzburg has a restaurant that focuses on organic products (including Hirter organic "bio" beer, which is quite good) and it has several vegetarian options. My mother loved their cream of asparagus soup (made with vegetable stock, not chicken).
Vienna is far from being just wienerschnitzel and sausages. One of my favorite restaurants in Vienna is Hotel Riviera (not a hotel at all, just a restaurant) tucked away on a small side street behind St. Stephan's cathedral. A mostly local crowd, very fresh food, pleasant decor and ambience. You can view the menu online at their web site, It has many vegetarian options but is not a vegetarian-only restaurant. Cash only, as I recall.
http://www.lametta.com/
also: http://www.wien.info/article.asp?IDArticle=13751

Wrenkh, of course, is the most famous vegetarian restaurant in Vienna; it is mostly if not 100% vegetarian (you may find the odd bit of fish on the menu).
http://www.wrenkh.at/wrenkhneu2/atmos.php

I enjoyed casual but tasty meal at Country Life in Prague. Can't remember the exact address, but it's in Old City, very conveniently located. They also have health food stores with some take-away items.

There is a veggie place called Vegetarium in Budapest. Vegetarian version of Hungarian cuisine. Loved the potato pancakes and goulash. Very comfortable seating.

Not sure about Vienna but I would probably gladly survive on kaffe und kuchen!