- I had an old version of this board lying around for some time. Never got it to work nicely with the old Arduino IDEs...
As it was one of the first boards, it still had an outdated bootloader on board. Flashed now with the 1.5.2-IDE the new Leonardo-bootloader and encountered the same problem you are describing here: TX and RX pins are always "on". As these guys are quite bright on that board, that's kind of annoying...
Well, the solution turned out to be simple: change the board type in the Arduino IDE to "Micro" (just below the Leonardo-listing). The first time you upload code, the board will reconnect again, your OS (I have done this under WinXP) will detect a "Micro", requesting to install new drivers. Do this. This will get the TX and RX leds into the right mode, i.e. only signalling traffic, not constantly "On".
Note that the serial port the device is attached to will change during this procedure. Check your device manager for the new port and set it correctly for further uploads.
So in summary: seems that the IDE has some issues here - when the IDE thinks it talks to a "Micro", everything is ok. When it thinks it talks to a "Leonardo", the TX and RX LEDs of the Olimexino stay always on.
I have not checked whether the Leonardo bootloader is different from the Micro bootloader and what the IDE is doing differently. But with my board has still having the Leonardo bootloader "on board", selecting "Micro" as board works fine.