It's really not about the shoe brand, it's about your feet.
I'd start with some of the known 'good brands' of walking shoes such as Nike or New Balance (there are others out there, I'm sure more will be posted in this thread) and look at shoes in your particular shoe type.
Type meaning...do you have normal arches, high arches or low arches? Do you walk splay-footed, pigeon-toed or keep your toes pointed straight? Is this behavior in one foot or both? These questions are very important in narrowing down the type of shoe you need to look at. For example, I have low arches and keep my toes pointed straight, so I tried on all the shoes that fit into those needs and settled on a pair of Nikes.
Have you ever had your feet professionally measured? If so and it has been over 6 months (3 months if you have lost or gained a large amount of weight in that time period), have them measured again. Many people wear the wrong size shoes because...that's the size they've always worn (wrong!).
This podiatrist has a wonderful short article about 'happy feet':
http://www.surreychiropodist.co.uk/happyfeet/