Odorheiu makes an ideal base for visiting the many small fortified village churches in the area - for example Darjiu (with hams and grain still stored in the walls), Bogoz (which has a fresco of St Lazlo), or Enlaka (with ancient runes). You can get there by car, or even on a bike, and a taxi isn't too expensive. (Firms have fixed rates according to distance, and the drivers usually speak a bit of English.) There are buses too - the tourist office in the square has timetables. When you visit a church, you're normally expected to donate a little money.
You can also visit Cristuru Secuiesc, where Sandor Petofi spent his last night, and see the tree under which he recited his last poem.
Roads are generally pretty good, the exception being the road to Enlaka, which is appalling.