Two new classes of exact solutions in vacuum Brans-Dicke theory are obtained, each of which is a two-way traversable wormhole for the coupling parameter omega<-2 or -2<omegaless than or equal to0, respectively. Each of the two new classes of exact solutions satisfies not only the general constraints given by Morris and Thorne [Am. J. Phys. 56, 395 (1988)], as concluded earlier, but also the constraints from a trip through a wormhole. It also follows that the scalar field phi plays the role of exotic matter violating the weak energy condition.