Motor Driver - Microcontroller-less and no IC chips available
I always prefer to make a difference between "Power Driving," "Motor Controlling," and "Robot Decision Making" 


For power driving of a DC motor that needs to rotate in both directions, you need a circuit based on the H-Bridge concept. 


To make an H-Bridge you can have different solutions based on your requirements. If you need speed control you can use 4 transistors either BJTs of FETs are usable. 
If you just need direction control (at full speed), you can use 4 SPST relay in the place of transistors, or you can use 2 SPDT relays. Because either leg of an H-Bridge consists of a high-side driver and a low-side driver which never turn on at the same time, you can use an SPDT relay.
There is also a little trick that you can use 2 SPDT relays to control the direction and use just one low-side OR high-side transistor to control the speed. It's preferred to use a PWM signal to control the speed.

Controlling the direction is easy with a single binary signal, but, as I mention above, controlling the speed needs more complex signal generation which is a little difficult without a micro-controller. You can use some old methods like 555 to generate the signal. 
About decision making: You said that you need a 45 degree angle. It depends on the robot mechanics but I think you can use basic logic for that, something like flip-flops and basic gates, but in simple systems maybe a few transistors are also enough. As I said, it depends on the physics of it.