Yes, as long as you separate "taking up residency" from "getting a residence card": if they are planning to spend more than 90 days in any country, they are strictly speaking supposed to apply for a residence card in that country. Whether this will actually be possible depends on the country and on how much longer than 90 days they want to stay. In some cases, they might be told not to worry about the residence card if they're staying, for example, for 100 days.
If they want to spend no more than 90 days in any country, no residence card is necessary.
The consequences of failing to get a residence card will also depend on the country, but will in no case be worse than a fine. It's probably a good idea to weigh the negative consequences against the cost of preparing the residence card application.
In France, for example, there's no actual fine; instead, if the spouse is discovered in France after 90 days, she will be told to get a residence card, and the application fee will be €340 instead of €25. But if the couple is about to leave, the spouse won't actually need to apply for the card. This is how it works for my parents: When they leave, my mother is found to have spent more than 90 days in Schengen, and then she is seen to be the spouse of an EU citizen, and they stamp her out.
For a more specific answer, a more specific itinerary would be necessary, since the particulars depend on the country.
Note that the term is "residence card" rather than "residence permit" because it serves to document a right that has its source in EU law, not in a grant of permission from the immigration authorities. A consequence of that is that if their administrative procedures are not sufficiently agile to handle the case of a stay that's slightly longer than 90 days, they can't forbid you from staying for slightly longer than 90 days.