One possibility would be to use a turns feature class or table to model an increased time impedance for all turns.
The idea would be to use a high value for turns (such as 15-20) seconds, and a low impedance for straights (0-5).
Then, as routes are created, every time a turn is the "fastest route", the additional impedance time will likely influence the route to go straight. Of course, turns will most likely have to be made eventually, but at least this way the route will choose to make as little turns as possible because of the larger impedance on all turns. This page also some more information for how impedance is calculated for different network analyst products.