Problem driving 3V motors with TLC5940 and transistor

This doesn't turn on the motor but there's 0.36V voltage that drops across the motor.

If your motor takes 100mA at 3.7V then it has an effective impedance of 37R. If you only have 0.36V across it that implies you are only getting 10mA of drive from the chip. Are you sure you put 200 ohms in the IREF resistor? Have you used the software that lights the LED to be as bright as possible?

I am assuming you are measuring the voltage on a meter, this does not always show up voltage well that is being pulsed by PWM so the reading might be meaningless. You should look at it on a scope if you can. The other thing to try is putting the other end of the motor to +5V.

Where shall i start?

Look at the power formula in the data sheet. It might look intimidating at first but it just breaks down into two parts, a fixed part for the chip and a variable part for each of the loads you are driving.