Did ancient armies march while reciting [] poems?

Seemingly yes, and some amusing accounts of these poems survive, in sources such as Suetonius Lives of the Caesars.

Roman soldiers, belonging to specific legions, are recorded as having special songs, sometimes about their commanders. Suetonius (Divus Julius 51) accounts for one that sounds quite funny, supposedly sung during Julius' Gallic Triumph:

"‘Romans, lock your wives away: the bald seducer’s in the rear, You’ve squandered on his Gallic vice the gold you lent him here."

Alternative translation (by Janet Lembke):

Citizens, watch your wives: we bring you a hairless adulterer. He's spent all the fucking in Gaul; here he's taken out loans.

Others were (supposedly) less flattering:

"All the Gauls did Caesar vanquish, Nicomedes vanquished him; Lo ! now Caesar rides in triumph, victor over all the Gauls, Nicomedes does not triumph, who subdued the conqueror." (with reference to Caesar's supposed having-been penetrated by Nicomedes, now dismissed by contemporary scholarship as having been a vindictive rumour).

(Suetonius, Divus Julius, 49).

Suetonius, in the Divus Aurelianus (6.5), accounts for other soldiers' songs:

"We've cut off a thousand thousand thousand heads. One man has cut off a thousand heads. He'd drink by the thousand who killed by the thousand - No one has enough wine to match the blood that has spilled. A thousand Sarmatians, a thousand Franks - again and again we've killed them. Now we look for a thousand Persians" (it's much less wordy in Latin)

Further reading: Janet Lembke (1973): 'Bronze and Iron: Old Latin Poetry from Its Beginnings to 100 B.C.' (see esp. pp 49-50).