Without having read your stories, it's hard to say. There are lots of things one could do in a story that if done well can be clever, innovative, and effective, but if done poorly can come across as a gimmick, lame, and tiresome. In general I'd avoid doing something unusual just for the sake of doing something unusual. But as I say, if you do it well, it can be way cool.
Why do you want to not give the characters names? If you have a specific purpose, like you want to give the idea that the people around them don't know or care who they are, or you want to emphaisze their role over their personal identity, etc, this could be effective. If you're avoiding names to avoid preconceptions that a name might give, like you want to avoid giving the character a national identity or be vague about whether this person is male or female, maybe. But if you're just doing it because you think it's a clever gimmick, I think that would be hard to pull off for ten stories without it just becoming annoying to the reader.
If a character has a title or maybe some other handy "identification tag", this would be little different from a name in practice. I mean like, if you call a character "the mayor", "the sergeant", "the prisoner", etc., the reader might not even notice that you never give the character a name.
You could always use descriptive terms like those you mention, but I would think this could quickly sound like an affectation or a gimmick. If a character only appears briefly and is thus only referred to two or three times, this might again pass pretty much unnoticed. This is especially true if the character would be largely anonymous. Like, I know you said none of the characters in these stories have names, but if you had two or three main characters with names, and then you mention they go to a store and "the clerk" did this or that and then "the manager" said this, readers would probably take this as routine: we usually don't know or care about the names of store personnel, so calling a store clerk by name might be odder than just calling him "the clerk".
A long description could get tedious. If you call a character "the tall bearded man with the snakeskin boots", using that phrase over and over could be tiresome.
I'd be careful about having different descriptions of the same person. If here you call him "the accountant" and there you call him "the tall man", the reader may have trouble grasping that this is the same person. I recall a book I read recently where there was a character who was routinely called by name, but every now and then -- like once a chapter -- the author would suddenly refer to him as "the redhead". The first time he did this I was totally confused, as he had never before mentioned the color of the character's hair. I had to re-read a couple of paragraphs to figure out who he was talking about. Then when he did it again some time later, I knew who he was talking about, but it was distracting.
If you find yourself giving convoluted descriptions, like, "And then the second man -- not the man who had arrived in the bus that is, the other man -- said ...", I think you definitely need to back up and rethink.