The serial port of the Arduino can only sent bytes (value 0..255) over the line. The receiving application interpret these byte values - typically as ASCII, however you are free to use another interpretation including Greek characters. The Arduino does not know, it just send bytes.
It's similar to writing a word document, selecting all and change the font to Greek. The (underlying) bytes won't change, but the interpretation and visualization is changed.
Several LCD screens have the option to define your own characterset. If there are enough free definable chars in the LCD you could define the whole greek alphabet.
If you are sending the data to a PC/Mac, it is up to the receveing app to translate the byte to greek.
A workaround I think of is to place the Unicode (greek - subset) characters in EEPROM and overload the print statement in such way that instead of [byte] it will send EEPROM[byte] and EEPROM[byte+1]. However the receiving app must expect and understand Unicode. As there are 512 bytes in EEPROM there are just enough memory places to do this.
Another workaround is to define two special characters to shift forth and back a fontset. This technique was allready used in Morsecode. If the '>' is sent the next fontset is used and if '<' is sent the previous fontset is used. The receiving app must understand this protocol and it probably only will do this if you write it yourself.
Hopes this helps.