Lustique,

Not sure how you are searching but I think the key is to establish a rapport and some form of communication with the landlord. I think individual owners would be more flexible Vs. companies that are in the business of renting apartments. Sites id look for individuals that rent their apartments are:
viewit.ca
craigslist.org

Shoot back with any questions. I arrived here a month back, dont have a CDN credit card (have a US one that I still use) and dont have a D. License, no job yet, zero credit history as they dont look at the US credit bureaus in most cases but (touchwood) I have been able to rent an apartment. My bank has applied for a cr card for me but its in the process as well.

Some may require you to pay an additional security deposit since you dont have a credit history (refundable id say).