Warmbloods do seem to rule the dressage ring, but I have seen a variety of horses do well despite being breeds that are not generally seen as dressage horses (Quarter Horses, Draft breeds etc). Generations of selective breeding have gone into making Wambloods ideal for competitive dressage, but I have met a few Warmbloods that have not been great as dressage horses (but loved jumping). I would look for a breed that piques your interest in terms of temperament, appearance etc... and then look for individuals that display the talent / movement that you need for the dressage ring.
When I was looking for my dressage partner I started by looking for a horse whom I would enjoy as a companion and mount first and foremost... when you find your ideal equine friend it is a lot easier to put the work into developing your horse's technical skills, because you enjoy being together and working with him and you want to do so frequently. ;)
I love my Baroque-looking little Canadian Horse, and I look forward to when he is ready to show off in the dressage ring. He may not be a Warmblood, but I think that he will turn heads someday... of course I am quite biased! *lol*
If you like Morgans then you should also look at Canadian Horses - Canadian mares were used in the development of the Morgan breed. DNA comparison has shown the two breeds to be closely related, and if one looks at horses of each breed, strong similarities can be seen:
One of my childhood mounts was a Morgan mare - she was a great horse. Morgans are a nice, versatile breed. I am now quite smitten with the Canadians though (also a nice versatile breed)! :)
Any breed I think can do the low level dressage, some horses have more of the movment and athelticism which helps their chances of being able to go into higher levels.