Offer to take out insurance especially designed for pet owning tenants. The net has companies on it which provide this, do a bit of Googling and see what you think.
Offer a higher deposit or a slightly higher monthly rent.
Guarantee in writing that you'll pay for damages and carpet/furnishing cleaning.
Get references from, for example, a dog walker or sitter if you use one or, if you've taken on a dog from a rescue ask the homechecker if they will act as referee to prove your current home is in good, non-chewed, non flea ridden repair.
Advertise! Look at private landlords. Put an ad in Gumtree (be prepared for some dodgy responses, not all "landlords" on there are genuine so keep your wits about you - if it looks to good to be true it might be).
Put ads in shop windows in the area/s that you want to rent. Ask in local independent shops if the staff know of any properties going.
Be persistent with agents. In the experience of many friends agents who have read or assume no pets will just fob you off with that answer. Insist that they actually ask the landlord, explain things which are to your advantage (eg the dog is small/old/you've had him for 6 years/you're a SAHM and dog will rarely be left for more than a couple of hours/he's got his obedience certificates/he attends training or agility and is not left bored and destructive).
There's a website which offers advice on this, hang on and I'll find it (my friend had this dilemma a few months ago, she found a private landlord through posting in the free ads)