"The Middle Ages were a very interesting part of history" or "The MA was a very interesting.."??
I vote for 'was'. 'A very interesting part' = 'a very interesting period'. The term 'Middle Ages' is used here as a unit (span) of time, not an aggregate of multiple times.
Very interesting. I was perplexed, and the agreement of several dictionaries does not resolve my perplexity. It seems to be a single historical period, yet considered a plural noun:

Webster's: Middle Ages (noun plural) the period of European history from about A.D. 500 to about 1500.

American Heritage:
PLURAL NOUN: The period in European history between antiquity and the Renaissance, often dated from a.d. 476 to 1453.

Compact Oxford:

plural noun the period of European history from the fall of the Roman Empire in the West (5th century) to the fall of Constantinople (1453), or, more narrowly, from c.1000 to 1453.

When in doubt, follow precedent, I suppose: The Middle Ages are an interesting period of European history.

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Does a plural noun always take a plural verb?

To me, the term sounds like it deserves, in some circumstances, a plural form in the verb, and in others, a singular.

The particular quoted sentence here seems to call for the singular because of the term 'part of'.

'The Middle Ages were full of important changes in philosophy, art, and literature.'

'The Middle Ages was a critical period in European history.'
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I'm just quoting, Dave. I remain impaled on the horns of the dilemma. I myself would taste each instance and see how it lies on the tongue. I like your examples.