Alles Wichtiges steht in der Frage da oben, aber ich kann es auch ein bisschen erläutern.
In English when I'm describing a period of time in the past I usually say either,
"At this time...", "During this time...", or "In this time..."
When I think about finding equivalents for these phrases in German "Während dieser Zeit..." and "In dieser Zeit..." seem to cover the last two respectively.
As far as the first phrase goes, I have often noticed "Zu dieser Zeit..." being used in a manner that seems to give a rough equivalent. Is this correct to assume? Are there any nuances of meaning that "Zu dieser Zeit..." has which aren't covered by "At this time..."?