Is there anyother way to construct an FM transmitter
I made this circuit before and it works properly. Here are the issues I met, they might be useful for you:

1- Don't use a battery lower than 9v. You can use a higher voltage.

2- Using a very long antenna will help you a lot. If you don't have a long wire, you can connect an aluminium pot :)

3- The frequency is not stable. so, Each time you turn on the circuit, you should tune your receiver (Change the frequency of your receiver) until you get the best result.

4- You should Not use a Mic as "Audio In". And I think it's better to rise the volume (Volume up) of the input audio.

5- The Capacitor C3 is important.

6- Don't worry about the shape or the number of turns of L1. Just bring a pen and turn a wire around it 5 or 8 times. and If you need to change the transmitter frequency, Change the values of C4 and C5.

7- I actually used a lower value of R3. I used 100 ohm instead of 470 ohm.

8- After all of the previous steps. If it still does not work, Change the value of R2 using a potentiometer.

9- My transistor part number is S9014. I did not use 2N3904 because I don't have it. 

If you need another circuit, the following circuit is easier to construct and it works too: