the smaller part (the post) is connected to positive
Almost always true, but alas, not quite always. There are a couple of odd LEDs where things are the other way around, and LEDs with multiple chips (bi-color or tri-color) and a common anode or cathode will most likely have the "common" signal be the larger "anvil" part.
When in doubt, I use a small coin cell (2016 or 2032) to test the LED.
It sounds like you may have put the LEDs in consistently backward. For freeduino, they should all have the cathode (usually the flat side) toward the top of the board. If they are in backward, it won't affect the rest of the board - they're only indicators and aren't necessary for actual operation (you can leave them out to save money, if you start with a bare board!)
Also, since they're nicely current-limited, you PROBABLY haven't damaged the LEDs either.
And the risk of damaging the board as a whole by reversing the LEDs is probably minimal too.
You CAN test your board without removing the LEDs first. Assuming that the LEDs have a flat side, and that they're all installed with the flat side pointing the same direction, you can get a good hint by testing only one of the LEDs, and the most easily accessible is the pin13 led. If you remove the Atmega chip (and leave the Freeduino otherwise unpowered), the pin13 LED will have one side connected to pin13, and the other side connected to GND (immediately adjacent to pin13 on the connector!) Use a small 3V battery and a couple wires. If the LED is backwards, it will light when the + side of the battery is connected to GND and the - side is connected to pin 13. Like in the picture here: