There are two reactions I get when talking to people about vegan travel. One is that the person is vegan, vegetarian or wants to learn more about how to travel and be vegan.
That reaction is actually more common that I would have expected. The other is that the person asks how I feel about missing out on so much of the culture by not eating meat or cheese.
Granted, a lot of regional specialties are meat- and/or cheese-centric, particularly in colder areas, but there are also a lot of interesting vegetable dishes. Most cultures have historically not eaten the amount of meat currently eaten in North America, so a lot of their traditions are actually based on plant foods.
In restaurants you won't see much of these dishes, since they prepare the ones that are impressive and expensive, but if you do a little research you can find so many that you can make yourself. Another excellent vegan travel strategy is to stay at a bed and breakfast or pension where the owner makes home-cooked meals, and are usually happy to make you a vegan meal if you explain what that means.
The idea that food is the most important aspect of experiencing another culture is very limited. While it is interesting and fun, it is definitely not the only way to experience life. Vegan travel does limit your restaurant choices, but it doesn't limit your choices for museums, meeting locals, going to concerts, plays or sports events, or so many other cultural activities. If food were the only interesting thing to do when traveling, it probably wouldn't be as popular as it is.
A great way to get the real feel of a place, and enjoy an easy vegan travel meal, is to have a picnic. You can get some of the local foods, fresh produce or just grab some takeout. Then find yourself a bench or a patch of grass somewhere central, and enjoy the view and/or people watch.
You can temporarily check out of looking like a tourist and pretend you're a local. Some of my most memorable picnic locations have been under the Eiffel Tower, on the beach on the south coast of Turkey, and in a corner of the Medieval wall surrounding Rhodes.
Meeting up with local vegans is a great way to both learn the best vegan travel options, as well as meet new friends that will show you a side of your destination you probably otherwise wouldn't see. You know that you already have one thing in common, and often the mindset means there are more synergies. Couchsurfing.org is perfect for connecting (and can find you a place to sleep), and there are lots of other great websites.
Although vegan travel doesn't include meat, eggs or dairy, it can most certainly include a lot of culture and a lot of fun. Don't let yourself feel limited and you might find that by learning how to travel and be vegan, you actually have a richer experience than you would have otherwise.