I am making a druid and I'm trying to decide between having a riding dog and a wolf as an animal companion. Can dwarves ride riding dogs? That could be the deciding factor.
No. Dwarves and riding dogs are the same size, therefore are not compatible.
According to Rules of the Game:
A mount must be at least one size category bigger than the rider.
This is also on page 204 of the Dungeon Master's Guide:
At least one size category larger than the character. Also, a flying mount can carry no more than a light load aloft.
Riding Dogs are Size/Type: Medium/Animal
Medium load capacity for a riding dog is 100–200 lbs., and over 200 is a heavy load. The Player's Handbook says dwarves are slightly lighter than humans which range between 125-250. Think Gimli from the Lord of the Rings on a St. Bernard. Doubtful he could carry you very fast or very far.