The simplest solution is to buy LEDs in the colour of your choice that are described as 5v LEDs, otherwise you need to add a resistor in series. If you do neither any standard led will blow nearly instantly.
For 5V your most LEDs should light. There are three issues that could cause the LED not to light.
1. The LED is connected in reverse.
2. You have exceeded the max current of the LED and it is now fried.
3. The LEDs were bad out of the box (unlikely).
While forward voltage is important to get the LED to light. Probably the most important rating is the max current of the LED. The purpose of the series resistor is to reduce the amount of current flowing through the LED. More advanced techniques would use a buck current regulator to maintain current more efficiently.
Put a 220 ohm resistor in series with the LED, connect that pair to the 5 volts, and measure the voltage across the LED. You will probably only have about 10 milliamps flowing, but it will get you a voltage number to calculate the real size of resistor it needs.