Theetu will not be for one year but ten days only. It is called deeksha, during that one-year period; couples who are karthas should not go on a pilgrimage and take holy dip in sacred rivers. But if we happen visit a place where per chance a holy river runs through, then taking bath in that river is permitted.
For example, if we have to visit Srirangam for some reason, then taking bath in Cauvery is considered special, similarly in Ganges as well.
Going on a pilgrimage involves taking an oath to visit sacred places and bathing in sacred waters so pilgrimage is not permitted during that period, otherwise when one gets an opportunity by chance to bathe in holy river, it is considered fortunate.