Should we adopt the morals of our time without question?

If we adopt the morals of our time and act accordingly, we will be regarded as moral through our actions, which align with what is considered good and moral. Plato has something interesting to say in this respect, described in the book that I'm currently reading: A brief history of Ethics. He sort of answers this question as he makes explicit his rejection of 'rhetoric' as a morally neutral action. 

Here I describe very lightly some of the historical framework that birthed this rejection to begin with.

The Greek Agora was typically a square in Ancient Greek cities that served to their pondering, law-making, philosophising, intellectualising and socialising. It was largely more culturally common then to be an active participant in political and philosophical discussion than it is today. It is easy to imagine that one's ability to persuade one another and have the winning argument would be prized highly. Ancient greeks also experienced a heavy crossroads of cultures, including among themselves. If city A values X, and city B values Y, and townsfolk C value Z, then one can choose to live at any one place with the morals that one finds most preferable. No one actually does this, but crux is the realisation that the average greek thinker has that morals are not ubiquitous or universal. It becomes inevitable to be confused and detached from one's own ethics.

The sophists were teachers in Ancient Greece, among other things they taught Rhetoric: the skill or virtue to persuade. Plato disliked it, considered it deceptive, considered it not morally neutral. Now that's the interesting bit. Rhetoric, persuasion, is not the morally neutral action that it pretends to be or is assumed to be even to this day. This can be explained by developing that statement. 

In order to hold that such techniques are neutral, it is necessary to also hold that it is morally irrelevant whether a man comes to a given belief by reasoning or in some non-traditional way[1]

Then it also follows that one is not responsible as a moral agent, for the path which he took to arrive to it is irrelevant, because of the disregard of the intentionality of his actions. It is just as moral to take 'good' action voluntarily as it is because of blind predisposition.

I would argue that it is intentionality the very object upon which morality exerts itself, and a model that ignores it in its moral framework is very deeply flawed.

And therefore, to answer the title, the way in which we arrive at our moral code is not irrelevant or a-moral. It is very much moral or immoral. 

[1] A short history of ethics, Alasdair MacIntyre, pp 26, https://www.goodreads.com/book/show/1871651.A_Short_History_of_Ethics