Do landlords accept higher security deposits in place of a strong credit history?

[Jay C]

I am searching for my first apartment and completely financially capable of paying rent. Unfortunately, my credit history is pretty unestablished because I never used credit cards. (Prefer to pay cash...silly me.) Anyway, does this mean landlords are going to slam the door in my face, OR, will they let me give them a higher security deposit to compensate for my lack of credit history?

Any advice would be very, very much appreciated. Thanks!