I've been going through some of R. Elchanan Samet's shiurim on Tehilim (in Hebrew; the English ones are coming out a bit slower). I might have mentioned it before, but I find R. Samet to be one of the most exciting contemporary commentators on the TaNakh. I've read a lot of his stuff - on Chumash, on Melakhim, and now on Tehillim, and he's really very good.
In the shiur that I'm currently reading, he analyzes Psalm 44. He often takes the descriptions of events recorded in the psalms and tries to link them with specific historical events. With this one, he runs into trouble. He presumes that the canon was closed, or at least that no new psalms were added to the collection that forms the Masoretic Book of Psalms, from some point during the Second Temple. Thus, this psalm, which refers to exile, must refer to teh Babylonian exile. This, however, presents three problems:
a) The psalm makes no reference to the destruction of Jerusalem or the Temple, which would be expected of a complaint about the Babylonian exile.
b) The psalmist protests the innocence of the Israelites (or the Jews, at this point); this flies in the face of everything else that the prophets wrote - that the exile was a punishment!
c) The psalmist very stirringly relates to the fact that the Jews went to the slaughter for God's sake. This is not characteristic of the Babylonian exile - which seems to have been quite comfortable.
These are R. Samet's questions.
I believe that I can point to a set of biblical events that would answer the questions.
As is familiar to students of TaNakh, the Jewish King Yehoyakhin was exiled to Babylonia about a decade prior to the destruction of Jerusalem and the Temple. Along with the king, a group of several thousand of Judea's most promising youths were exiled as well.
So there's your answer. The psalm does not lament the destruction, because it had not yet occurred. It protests innocence, because these youths were, in fact, innocent. It relates to martyrdom because the first biblical account of martyrdom - that of Chananiah, Mishael, and Azariah - is actually from that time period.