Do not, under any circumstances, allow your horse to be 'blessed' by a wayshrine. It's only possible to do if you get your horse on top of it and then make sure it is between you and the shrine when you activate it. However (and I read this on an Oblivion wikia) doing so not only cures your horse of any diseases it may pick up on the road (from attacking animals and such, same ones your character can pick up), it also gives your horse an insane boost in speed.
I know this because I just put Prior Maborel's Paint on one to see what happened. I thought the warning was maybe a tad overstated, but it's not. In run mode, the horse is so insanely fast that it literally runs off cliffs before I can turn it. In walk it's probably as fast, or faster, than the black horse.
So, if a wayshrine can make the paint (the slowest horse in the game) so insanely fast that it can't be controlled, just imagine what it can do to the black (the fastest horse in the game). Oh, and the effect is permanent, not like the blessing if it hits you.
Therefore, if you want to be sure of keeping your horse free of diseases, my suggestion is to get to a spellmaking altar (either the Arcane University or Frostcrag Spire) and make a custom cure disease spell that works on touch instead of on self. Naturally, you'll need to go to a chapel somewhere and buy it from the spell vendor, and then get your restoration to at least 50, but then you can just cast it on your horse from time to time to keep it healthy and disease-free. The reason I'm saying this is that you can't tell when your horse has been afflicted with a disease, unless you notice it isn't healing as quickly as it should. Therefore just hit it with a cure disease once in awhile.
While I'm on the subject of your horse, you should also make yourself a custom restore health spell for your horse as well (just make sure it's on touch and not on target or self). If you get your restoration up to 75, you can make one that restores 100 points of health for 1 second. Yeah, it costs a lot of magicka to cast, but it will be worth it after you find your horse to be down to very little health after it's attacked by a minotaur or an ogre.
So, happy hunting in Obilivion.