So without deodorant, lack of regular washes and no sophisticated waste disposal system, how much of an issue was bad odors, especially in public places across ancient cultures? Was it something people just put up with and got used to or is there record of complaints against it?
The problems you mention do not apply only to ancient societies, but pretty much every large city before the 20th century. For instance, while internal combustion engines cause a great deal of pollution and smoke, it is still far preferable than having piles of horse dung pile up in alleys, as was the case in New York, Paris, London, etc.
http://www.uctc.net/access/30/Access%2030%20-%2002%20-%20Horse%20Power.pdf
In these cases, people do what they have always done. As long as the smell isn't bad enough to force an involuntary reaction such as retching, people find a way to deal with it. Think about the people today who work at dairies or perform autopsies.
The tradition of scenting oneself to cover bad odours is pretty old, but probably wasn't as widespread in the past as is it now.
There is some evidence of the ancient egyptians having developed a deodorant in the slightly controversial hearst papyri and quite a few roman sources mention gifts of perfume and also [complained] (http://en.wikipedia.org/wiki/Cosmetics_in_Ancient_Rome#Perfume) quite a bit about people who put on too much.
Fun fact, the in 17th century England, people liked lawns to smell nice, so people could 'wash' the smell of the streets off their boots.
In general, the populations around the mediteranean were very concious of the sort of smells that happen when your vehecle needs to excrete
There is a very good book called the [perfume](http://en.wikipedia.org/wiki/Perfume_(novel) that explores what the world of late renaissance france smelled like.
There are some great responses here about cities, but remember that most people before the last century lived on farms. Those farms would have smelled a lot like a modern small farm that raises livestock today - lots of strong smells, but nothing you can't get used to (and nothing like a city full of a million of people's shit decomposing in the summer heat, as /u/AlyoChem describes for 19th century London - that's quite a pungent picture, but thankfully not the norm for most humans throughout history!). Visit a modern dairy farm and you'll get a good idea of what things smelled like for most of our past.
And people's bodies in the past definitely smelled stronger than modern Americans, though many cultures (certainly most in the middle ages, which I study) practiced some sort of regular bathing. And if you've ever been camping, you may have noticed how if you stand in wood smoke, body odor tends to get masked by the smoke's scent until everything smells smokey? That had to have been common in the past, especially in places like the houses and halls people lived in in the early middle ages, whose hearths didn't have chimneys and which were filled with a lot of smoke. There would have been a lot of BO in those halls, but smoke would be the dominant flavor. So again, the average farmer probably wouldn't have been constantly disgusted by the people s/he was smelling. But s/he still would have been exposed to a much wider range of smells than we are in our day to day lives.
Also bear in mind that strong, bad smells might not have been interpreted the same way by people in the past as we do in the modern first world. For example: there's a palace complex in Lyminge, Kent, that the local kings used in the seventh century, and which has been excavated over the past three years. This summer, they found a giant midden spread (trash dump) about 60 meters from the palace. This dump was being used at about the same time as the palace, and contained a lot of animal bones and other trash from feasts in the hall. It had turned into beautiful compost, and must have been both a great testimony of all the feasts the king held (look/smell how many cows we ate!), and really good soil to spread around to help the crops grow. The fact that this midden was so close to the palace speaks perhaps of the servants'/slaves' laziness to not carry the trash any farther, but also really speaks to the fact that, for the kings of Kent, a pile of compost could be something to show off as proof of your ability to ensure fertility and a food harvest, rather than just a foul-smelling eyesore to hide (as a modern American might be inclined to do). It must have smelled strongly but, as my father always used to say when the wind brought the scent of a local dairy farm into our back yard, 'that's the smell of money!' You only get stench like that if you have a healthy, productive heard of cattle or, in the case of the kings of Kent, a lot of really fab parties. That makes it a good smell, even if you have to wrinkle up your nose a bit.