I know this is an old topic, but I gotta chime in. Of course this is always going to depend on the dog and the rider. I am very comfortable on my bicycle. I ride a lot. My dog is not super trained or anything, but I take him around by bike because it lets us go places faster and he likes the exercise.
It works fine for me. My dad tried it and fell twice and was pretty angry, so YMMV.
I use about a 1.5 m leash that I loop over the seat post. I ride along at around 10 mph mostly, although he always starts off with a 20 mph sprint out the door. He stays on my right. He knows not to run in front of the bike, I don't know how, but I guess it's common dog sense. Sometimes he pulls to the side to sniff something or if he sees birds or squirrels. If I think he wants needs to go to the bathroom I will slow down and pull over, or if not I will pull back left. I don't have a lot of leash out so he can't jerk on the bike with momentum, just tug on the leash a bit. He seemed to automatically understand that walking into the bicycle would be bad, and he never gets too close to the wheels. I don't make sharp right turns towards him.
I have never been pulled off the bike by him. Frankly, I don't see how it is possible. He is 50 lbs, but I have enough control of the bike that I can just lean the opposite way that he pulls, even if he sees a dropped slice of pizza that he really wants. The only accidents we've had is when first starting out he went to the wrong side of a telephone pole at low speed. The result was that I fell, and he probably got a bad pull on his collar, but now he is very careful to never go around the far side of any objects. He is also a little bit freaked whenever the bicycle moves around unpredictably, like if I am carrying it down the stairs.
We stick to quiet roads. If I ever have to pull into the middle (like for a right hand turn lane), I reach back to the leash and pull him closer to me so that he won't get run over by inattentive drivers passing on the right. I do the same if I think a parked car is going to pull out beside me.
Our longest ride was 10 miles, and he still wanted to run around the park after that. I don't know how I would get him the same amount of exercise without a bike.
Oh and here is a warning: Don't attach a leash to the handlebars, or attach a leash to your hand and hold on to the handlebars. I tried this the first time out and came very close to crashing.
Another alternative is to look at a bikejooring setup where the dog pulls you along. I think it would take more training though.