Thomas Adams nailed it I think. Softer soled shoes will work fine if paired to appropriate pedals. Where I've found soft soled shoes let me down is on long distance rides with pedals designed for cleated cycling shoes. The combination of limited contact area between pedal and shoe, being constrained to one position on the pedal by the clips and the soft sole creates "hot spots" and discomfort after a while. Unhappy feet. I can get away with it around town just fine, but on longer rides it's an issue.
I use my bikes mostly nowadays as daily transportation in lieu of driving, so I've got no use for stiff soled cycling-specific shoes. The discomfort walking in them is worse than their upside riding for me.
As for Nick's observations on clips and straps and the replies stating that one needs the clips and straps to hold one's feet in place on the pedals during bumpy rides, I was in Italy a couple of years ago with the use of an old Peugeot with Lyotard double-sided touring pedals sans clips and straps and the house being at the top of a potholed gravelly strada bianca of the usual Tuscan sort and as I hadn't ridden without clips in decades I was terrified my feet were going to slip off the pedals, resulting in a crotch toptube collision. Well you know what? After a week or so the fear was gone and I just rode down the hill and into town clipless and strapless over bumps and through potholes without even thinking about the lack of clips and straps, like when I was a kid on my Schwinn going over jumps in the vacant lot or like BMX riders do.
I'm seriously thinking about pitching the Campagnolo SLs on my main daily on-topic ride, finding a nice pair of BMX pedals in 9/16" and going without the clips and straps. Double-sided is really nice in town. Really nice. If I go that way I doubt I'll miss flipping the pedals and fussing with getting into the clips leaving stoplights one bit.
I think there's a divide between people who think of their bikes primarily as fitness/recreation devices who eat up all the bicycle specific gear like cycling shoes, clipless pedals, lycra shorts without pockets, dayglo cycling tops etc. and those for whom their bikes are actual practical transportation devices useful for moving oneself and a reasonable amount of baggage from one place to another and for whom affixing a bike to a bike rack on a car and driving it somewhere seems decidely odd and perhaps even illogical.