For those of us who don't use clipless pedals on our roadbikes, it would seem that the issue of cycling shoes is simple: Just ride in whatever shoes you like. However, at least for me it doesn't really work that way. The shoe has to feel right to cycle in, or I am miserable on a long ride.
The most obvious factor I look for is sole stiffness: My feet must be able to push the pedals without the soles of the shoes bending over the edge; otherwise it feels weird and tiring over the course of a strenuous ride. This is why most types of sneakers and casual slip-ons don't work; they are too flexible. The shoe must also be made of a breathable, yet waterproof material. It is uncomfortable to cycle in soaked cloth shoes in the rain. It is also uncomfortable to cycle in leather dress shoes, because my feet get too hot. Finally, the shoes should not be bulky, so as to fit into my Power Grips and generally not take up more space than necessary on the pedals.
I've been wearing Adidas running shoes on the bike, which have considerably stiffer soles than other sneakers and are also less bulky. They've been working well, except for one thing: the shoelaces. I tie the laces as securely as I can, then tuck them in under the "vamp" of the shoe to keep them away from the drivetrain. Mostly they stay there, but once in a while they come out and then I have to either stop the bike to fix them or risk getting them caught. And I have gotten the laces caught on the crankset a couple of times, so I am paranoid about it. Particularly on my fixed gear, the cranks are very narrowly placed and my feet are close to the drivetrain. I really do not want to get anything caught in there, and having to constantly watch for my shoelaces coming loose is annoying. So I reverted to childhood and bought velcro shoes. I've actually managed to find a pair of Adidas Originals, a special edition made with velcro instead of laces. Same stiff sole and breathability only nothing seems in danger of getting caught in the drivetrain. They look kind of silly, but as long as they work for me I don't care.
What shoes do you wear on the bike if you don't ride clipless?