I don't know Time::Piece; I've been using Date::Calc (functional UI) and DateTime (OO UI) for date/time calculations: the former has a bunch of Day_of_Week* functions, I'm not really sure about the latter, but I'm confident it has some suitable method too.

In particular:

#!/usr/bin/perl -l

use strict;
use warnings;
use Date::Calc qw/:all/;

print(Day_of_Week_to_Text(Day_of_Week(Today)));

__END__
[download]

Now referring to Date::Calc (with which I'm more familiar than with DateTime), you could get each date in the range of your Time::Piece object and cast it into a form that could be passed to a suitable function of the former module, e.g. a string in ISO format.

HTH