If the cheque is drawn on an account abroad you can still lodge it, the exchange rate you get will usually be slightly different than the Note rate, but it should be displayed. You can certainly ask them the rate they'll give you.
You will in all liklihood pay transaction charges as well. It takes a while to clear as mentioned above.
A Draft should clear almost immediately. If possible ask foreigners to send you a Euro Draft, or even a Sterling Draft.
In a former life I used to get Dollar cheques sent to me from the US. I used to lodge them in the Credit Union. I can't even remember what kind of value I was getting.
-Rd