Using OpAmp to drive LEDs according to incoming signal
R1 is already the current limiting resistor. Since no current is flowing into the opamp input, and its inverting input is at ground due to the feedback loop, the current through R1 will flow directly into the diodes. Therefore the current flowing through the LEDs will be 
$$ I_D = V_{osc}(t)/R_1 = \frac {5V} {4.7k \Omega} sin( \omega t)\, .$$
The current amplitude through the LEDs is approximately 1mA. In the equation I have assumed sinusoidal oscillation.

It might depend on the LED specification, but I feel the LED current a little bit low.