My husband and I are going to Vienna this May. We enjoy nice restaurants, but we are used to fish or seafood for dinner - and on this forum people seem to talk mostly about snitzel and other red meat, when it comes to food. I realize that occasional red meat wouldn't kill me, and I have it from time to time, but if I have it for dinner, I can't sleep because I am not used to it and it feels heavy... I'd rather have salad and fish, and save room for chocolate desert... Could anyone recommend a restaurant or 2 that serve decent (not fried) fish dishes? Thanks...

Of course you can have fish in Vienna! Trout is quite popular, as is pike-perch and salmon.
When I was in Vienna in March, I had trout at Glacis Beisl in the Museum Quarter (but it was very bony and I didn't do the world's greatest job of pulling the bones out), a delicious seafood appetizer at Steierereck and almost ordered the fish (pike perch, I believe) at Zum Weissen Rauchfangkehrer--I saw some other diners having it and they seemed quite pleased. None of those dishes were fried. Last year, I had a beautiful grilled trout at Meinl am Graben--one of my best fish meals ever (this year I had steak). FWIW, I never order schnitzel in Vienna. There are so many other fine Austrian dishes to try.
Of course, there is always the NordSee restaurant chain which has some nonfried dishes, but that's fast food, not really what you have in mind.
Or, to take a break from Austrian cuisine, you could have sushi or a cooked fish meal at Unkai, the Japanese restaurant at the Grand Hotel (Japanese owned). Considered the best Japanese restaurant in Vienna. Their Sunday sushi brunch is very popular.

In my opinion, the best fish restaurant in Vienna- Kervansaray at Mahlerstrasse 9 in the first District.

Try - www.kervansaray.at - they have some information in english.

founded by the father of the owner of Do and Co.
Not inexpensive , but well worth the visit- even for shell fish.

Also Dubrovnik on Reimergasse, First district. Excelent fish dishes- a family restaurant with husband cooking, family serving.

There are also a lot of chicken dishes - you're not stuck just with red meat - although there is often a lot of pork and game.

Totally agree with Kervansaray recommendation and Japanese restaurants are very abundant in Vienna and for the most part (not Grand Hotel) are very good value. The Akakiko near Freyung was a reliable favourite (but not the other Akakiko next to Billa closest to Stephansplatz). I find the trout served outside summer period very bland - perhaps more farmed than in summer.

I had at least two dinners in Vienna featuring very good fish entrees. One was atlantic char and the other was red mullet. Both delicious.

Chocolate dessert - boy oh boy are you in the right city for that! Try any version of "moor im hemd" or "moor in a shirt", that is, chocolate pudding with a cream sauce or whipped cream topping. The best I had was at a place called Immervoll - it was like a piece of "slumped" chocolate cake with a billowy cloud of schlag (whipped cream) around it. I can't believe I ate it all but I did.