I actually enrolled into this forum because I had the same idea and was wondering if I could use Arduino for a similar project of mine.
Please note that I haven't yet read through all the Arduino documentation and am not familiar with the capabilities of all the different boards, but I do know some stuff that led me to the questions I'll be asking now, so feel free to bitchslap any mistakes I make... ;)
I had the idea to use capacitive sensing of the chess figures. It's fairly simple to implement on Arduino (native support, according to the Playground section, articles http://www.arduino.cc/playground/Main/CapSense and http://www.arduino.cc/playground/Code/CapacitiveSensor), and by varying the capacitance of each chess figure (12 values: pawn, knight, bishop, rook, king & queen x 2 for each colour) you could (I assume) measure which figure was on which sensor at any given time.
In case of glass figures, you could encase the base of the figures with different metal rings to achieve the capacitance changes.
In case of full metal figures you could even go further - the capacitance would change dramatically when a piece was being touched by a person (allowing for the program to detect a person wants to interact with that piece at that field, highlight other chess fields the player could legally move that piece to by lighting up LEDs under legal fields - something like a tutorial).
The only problem with this method - as Arduino allows (as far as I could gather) up to 6 capacitive sensors per Arduino, this means I would need 11 Arduinos just to do the sensing? :o This would be an overkill... Or am I wrong? Can the Mega deliver the punch? ::)
Or am I completely wrong and this isn't possible at all? :-/